#95231

order	512
nilpotency class	3
isoclinism	isoclinic class of #95192
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₄³	C₂³
prime radical	C₂⁴	#95
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

540 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=gh,d^2=gi,e^2=h,f^2=g,b^a=bd,c^a=cg,d^a=dg,f^a=fghi,d^b=dh,e^b=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2427);