#6910552

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2416	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1412 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=gi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6817748);