#95633

order	512
nilpotency class	3
isoclinism	isoclinic class of #94978
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	C₂²×C₄²	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

866 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=gh,d^2=gh,e^2=i,f^2=ghi,b^a=bd,d^a=dg,f^a=fgh,c^b=ci,e^b=eh

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, 2829);