#367165

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1319 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=di,g^c=gi

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, 274361);