#134991

order	512
nilpotency class	4
isoclinism	isoclinic class of #12465
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1677
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8646
prime radical	C₂×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

275 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 275 conjugacy classes of subgroups — too many to draw. Only its 96 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dh,d^2=gi,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,e^b=ei,d^c=di,e^c=eh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42187);