#7090761

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1575 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,29,24,30,22,28)(19,26,32,41,20,33,35,25)(21,27,37,43,38,44,39,45)(31,42,36,48,34,47,46,40);(9,10)(11,15)(12,14)(13,16)(17,19,24,20)(18,25,30,41)(21,36,38,46)(22,32,23,35)(26,29,33,28)(27,42,44,47)(31,39,34,37)(40,45,48,43);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,24,19)(18,26,30,33)(21,34,38,31)(22,32,23,35)(25,29,41,28)(27,42,44,47)(36,37,46,39)(40,43,48,45);(11,16)(13,15)(17,21,22,39,24,38,23,37)(18,27,28,45,30,44,29,43)(19,31,35,46,20,34,32,36)(25,40,33,47,41,48,26,42);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,22,24,23)(18,28,30,29)(19,32,20,35)(21,37,38,39)(25,26,41,33)(27,43,44,45)(31,46,34,36)(40,47,48,42)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gh,d^2=fi,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,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, 6997957);