#492533

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

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

normal subgroups only

This group has 471 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,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, #72225563, #72225584, …
maximal normal subgroup of	#72225394, #72225563, #72225584, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399729);