#95716

order	512
nilpotency class	3
isoclinism	isoclinic class of #95118
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#95
commutator, prime radical	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

516 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=hi,e^2=i,f^2=g,b^a=bd,c^a=ci,d^a=dg,f^a=fh,d^b=dh,e^b=eg

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2912);