#94253

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

127 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 2 generators on 40 points:

(1,2,3,6,4,5,7,8)(10,16)(11,21)(12,25)(13,23)(14,24)(17,18)(19,36)(20,22)(26,39)(27,37)(28,38)(29,30)(31,35)(32,34)(33,40);(2,5)(3,7)(9,10,12,17,27,32,11,22,24,35,39,30,23,29,38,31,15,20,28,34,13,18,26,16,14,19,21,33,37,40,25,36)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fgh,d^2=g,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, 1449);