#145154

order	512
nilpotency class	4
isoclinism	isoclinic class of #145154
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#10431
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#28249	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

324 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 74 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|c^2=dg,d^2=i,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gi,c^b=cf,d^b=di,e^b=ei,f^b=fh,h^b=hi,d^c=di,e^c=eg,f^c=fhi,g^c=gi,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52350);