#6907986

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907977
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	#37659	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1989 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,7)(4,6)(5,8)(9,10)(11,15)(12,14)(13,16)(17,18,23,27)(19,24,30,40)(20,31,33,41)(21,26,22,36)(25,35,42,32)(28,43,45,48)(29,38,34,39)(37,44,47,46);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,16)(14,15)(17,19)(18,24)(20,28)(21,34)(22,29)(23,30)(25,37)(26,39)(27,40)(31,43)(32,46)(33,45)(35,44)(36,38)(41,48)(42,47);(1,2)(3,5)(4,6)(7,8)(11,16)(13,15)(17,20,22,32,23,33,21,35)(18,25,26,31,27,42,36,41)(19,28,29,46,30,45,34,44)(24,37,39,43,40,47,38,48);(2,6)(5,7)(10,14)(13,15)(17,21,23,22)(18,26,27,36)(19,29,30,34)(20,32,33,35)(24,38,40,39)(25,41,42,31)(28,44,45,46)(37,43,47,48);(3,8)(5,7)(20,33)(25,42)(28,45)(31,41)(32,35)(37,47)(43,48)(44,46)

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^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=dfhi,e^c=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6815182);