#6841619

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1163 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=hi,d^2=fgi,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=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, 6748815);