#135995

order	512
nilpotency class	4
isoclinism	isoclinic class of #12117
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1595
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#8853
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

411 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 411 conjugacy classes of subgroups — too many to draw. Only its 84 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:

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

Polycyclic presentation:

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

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, 43191);