#7080733

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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	#25131	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1611 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=hi,d^2=fgi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=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, 6987929);