#7090890

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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1590 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)(17,18,21,26,22,27,34,40)(19,28,31,39,30,24,37,23)(20,25,36,44,35,43,45,48)(29,42,32,46,33,41,47,38);(9,10)(11,15)(12,14)(13,16)(17,19,22,30)(18,23,27,39)(20,32,35,47)(21,37,34,31)(24,40,28,26)(25,42,43,41)(29,45,33,36)(38,48,46,44);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,33)(21,31)(22,30)(23,26)(25,42)(27,28)(29,35)(32,45)(34,37)(36,47)(38,48)(39,40)(41,43)(44,46);(11,16)(13,15)(17,20,34,45,22,35,21,36)(18,25,40,48,27,43,26,44)(19,29,37,47,30,33,31,32)(23,38,24,41,39,46,28,42);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(19,30)(20,35)(23,39)(24,28)(25,43)(31,37)(36,45)(44,48)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gh,d^2=fi,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,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, 6998086);