#95869

order	512
nilpotency class	3
isoclinism	isoclinic class of #95218
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	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

566 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=ghi,d^2=gi,f^2=g,b^a=bd,c^a=ch,d^a=dg,f^a=fghi,c^b=cgh,d^b=dh,e^b=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3065);