#95278

order	512
nilpotency class	3
isoclinism	isoclinic class of #95192
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	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

528 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=h,d^2=gi,e^2=g,f^2=h,b^a=bd,c^a=cg,d^a=dg,f^a=fghi,c^b=cg,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, 2474);