#94254

order	512
nilpotency class	5
isoclinism	isoclinic class of #94206
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1380
commutator	C₂×C₁₆	C₄²
Frattini, cosocle	C₂×C₄×C₁₆	C₂²
prime radical	C₂³	#367
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

119 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 2 generators on 72 points:

(1,2,3,6,4,5,7,8)(9,10)(11,23,25,46)(12,29)(13,35)(14,37)(15,21)(16,41)(17,24,42,30)(18,32)(19,49)(20,40)(22,38)(26,62,60,44)(27,64,61,43)(28,66,57,55)(31,52)(33,51)(34,50)(36,68)(39,47)(45,69,72,59)(48,70)(53,63,71,58)(54,56,67,65);(2,5)(3,7)(9,11,13,26,14,27,33,24,38,57,70,58,31,56,34,69,15,25,36,60,39,61,12,30,16,28,32,63,40,65,49,59)(10,17,19,43,20,44,18,46,41,72,29,67,47,53,68,55,21,42,50,64,52,62,48,23,22,45,51,54,37,71,35,66)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fgh,d^2=gi,e^2=fgi,f^2=hi,h^2=i,b^a=bc,c^a=cf,e^a=efghi,f^a=fh,h^a=hi,c^b=ci,d^b=dg

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, 1450);