#94349

order	512
nilpotency class	5
isoclinism	isoclinic class of #7422
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1347
commutator	C₂×C₁₆	C₄²
Frattini, cosocle	C₂²×M₃₂	C₂²
socle	C₂	#7258
prime radical	C₂⁴	SD₃₂
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂²×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

222 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 68 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 — 2 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fh,d^2=i,f^2=hi,g^2=hi,h^2=i,b^a=bc,c^a=cf,e^a=efgh,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=ghi,h^b=hi,e^c=ei

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, 1545);