#94014

order	512
nilpotency class	4
isoclinism	isoclinic class of #93295
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1276
commutator	C₄²	C₄×C₈
Frattini, cosocle	C₄×M₃₂	C₂²
socle	C₂	#7149
prime radical	C₂³	#349
LCS	G ▷ C₄² ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ #346 ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₄×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

144 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fh,d^2=g,f^2=i,g^2=h,h^2=i,b^a=bc,c^a=cf,e^a=efhi,f^a=fi,c^b=ci,d^b=dh,g^b=gi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1210);