#134666

order	512
nilpotency class	4
isoclinism	isoclinic class of #12833
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1807
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8595
prime radical	C₂×#41	C₄⋊C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

315 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 315 conjugacy classes of subgroups — too many to draw. Only its 128 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72217272, #72217274, #72224949, …
maximal normal subgroup of	#72217272, #72217274, #72224949, …
maximal quotient of	C₂×#134666

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41862);