#492530

order	512
nilpotency class	4
isoclinism	isoclinic class of #492530
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	D₈²
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

479 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 479 conjugacy classes of subgroups — too many to draw. Only its 120 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2)(3,18)(4,24)(5,14)(6,13)(7,16)(8,15)(9,17)(10,21)(11,26)(12,27)(19,35)(20,56)(22,59)(23,50)(25,38)(28,48)(29,47)(30,45)(31,46)(32,39)(33,44)(34,43)(36,51)(37,58)(40,62)(41,61)(42,53)(49,52)(54,64)(55,63)(57,60);(1,3)(2,10)(4,25)(5,29)(6,21)(7,33)(8,22)(9,23)(11,38)(12,42)(13,44)(14,18)(15,47)(16,36)(17,37)(19,28)(20,26)(24,60)(27,54)(30,57)(31,58)(32,51)(34,55)(35,41)(39,50)(40,63)(43,56)(45,64)(46,59)(48,49)(52,62)(53,61);(1,4)(2,11)(3,19)(5,26)(6,30)(7,27)(8,34)(9,28)(10,35)(12,39)(13,24)(14,43)(15,40)(16,45)(17,41)(18,42)(20,51)(21,52)(22,54)(23,25)(29,53)(31,62)(32,61)(33,55)(36,63)(37,38)(44,49)(46,48)(47,64)(50,56)(57,58)(59,60);(1,4)(2,12)(3,20)(5,30)(6,26)(7,34)(8,27)(9,28)(10,35)(11,39)(13,45)(14,41)(15,48)(16,24)(17,43)(18,49)(19,51)(21,55)(22,57)(23,53)(25,29)(31,61)(32,62)(33,52)(36,64)(37,38)(40,46)(42,44)(47,63)(50,60)(54,58)(56,59)

Polycyclic presentation:

2^9|e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,g^a=gi,h^a=hi,c^b=ci,d^b=de,e^b=eg,g^b=gi,f^c=fh,h^c=hi,e^d=eg,g^d=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225394, #72225565
maximal normal subgroup of	#72225394, #72225565

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399726);