#7034395

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#42806	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1567 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)(3,6)(4,5)(7,8)(9,10,12,13)(11,14,16,15)(17,18,23,27)(19,24,22,36)(20,30,32,38)(21,26,34,43)(25,37,41,28)(29,39,35,48)(31,46,33,47)(40,45,42,44);(2,5)(3,7)(9,10)(11,15)(12,13)(14,16)(17,19,23,22)(18,24,27,36)(20,28,32,37)(21,29,34,35)(25,38,41,30)(26,39,43,48)(31,44,33,45)(40,47,42,46);(10,13)(11,16)(17,20,23,32)(18,25,27,41)(19,28,22,37)(21,33,34,31)(24,38,36,30)(26,42,43,40)(29,45,35,44)(39,46,48,47);(1,3,4,7)(2,6,5,8)(9,11,12,16)(10,14,13,15)(17,21)(18,26)(19,29)(20,31)(22,35)(23,34)(24,39)(25,40)(27,43)(28,44)(30,46)(32,33)(36,48)(37,45)(38,47)(41,42);(10,13)(14,15)(21,34)(26,43)(29,35)(31,33)(39,48)(40,42)(44,45)(46,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6941591);