#95016

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	C₄³	C₂³
prime radical	C₂⁴	#95
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

556 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)(11,19,22,23)(12,16)(13,17)(14,18)(15,21,24,20)(25,26,28,31,29,32,35,38)(27,30,34,37,33,36,39,40);(1,2,4,7,5,8,3,6)(9,11)(10,15,18,24)(12,20)(13,21)(14,22)(16,23,17,19)(25,27)(26,30)(28,34)(29,33)(31,37)(32,36)(35,39)(38,40);(9,12,14,13)(10,16,18,17)(11,20,22,21)(15,23,24,19)(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=g,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, 2212);