#95100

order	512
nilpotency class	3
isoclinism	isoclinic class of #94978
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	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

538 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)(11,17,19,14)(12,15)(13,16)(18,23,20,22)(21,24)(25,26,28,31,29,32,35,38)(27,30,34,37,33,36,39,40);(1,2,4,8,5,7,3,6)(9,11)(10,14)(12,20)(13,19)(15,23)(16,17)(18,21)(22,24)(25,27)(26,30)(28,34)(29,33)(31,37)(32,36)(35,39)(38,40);(2,7)(6,8)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(27,33)(30,36)(34,39)(37,40)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=h,d^2=gh,e^2=i,f^2=gh,b^a=bd,d^a=dg,f^a=fgh,c^b=cgi,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, 2296);