#135822

order	512
nilpotency class	4
isoclinism	isoclinic class of #2192
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#447
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8829
prime radical	C₂×#41	C₄⋊C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

294 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 114 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43018);