#96017

order	512
nilpotency class	3
isoclinism	isoclinic class of #95192
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

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

558 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=gh,d^2=gi,e^2=gh,f^2=g,b^a=bd,d^a=dg,f^a=fghi,c^b=ch,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, 3213);