#7034363

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#42883	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1596 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)(19,24,22,35)(20,30,32,37)(21,26,33,41)(25,36,40,28)(29,38,34,45)(31,43,44,47)(39,46,48,42);(2,5)(3,7)(9,10)(11,15)(12,13)(14,16)(17,19,23,22)(18,24,27,35)(20,28,32,36)(21,29,33,34)(25,37,40,30)(26,38,41,45)(31,42,44,46)(39,47,48,43);(10,13)(11,16)(17,20,23,32)(18,25,27,40)(19,28,22,36)(21,31,33,44)(24,37,35,30)(26,39,41,48)(29,42,34,46)(38,47,45,43);(1,3,4,7)(2,6,5,8)(9,11,12,16)(10,14,13,15)(17,21)(18,26)(19,29)(20,31)(22,34)(23,33)(24,38)(25,39)(27,41)(28,42)(30,43)(32,44)(35,45)(36,46)(37,47)(40,48);(10,13)(14,15)(21,33)(26,41)(29,34)(31,44)(38,45)(39,48)(42,46)(43,47)

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=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6941559);