#7063765

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	#42674	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1452 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)(11,13)(12,14)(15,16)(17,18,21,27,22,26,33,24)(19,23,30,34,28,36,42,37)(20,25,32,39,31,40,44,38)(29,35,43,45,41,47,48,46);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,23)(20,29)(21,30)(22,28)(24,37)(25,35)(26,36)(27,34)(31,41)(32,43)(33,42)(38,46)(39,45)(40,47)(44,48);(9,11,12,16)(10,13,14,15)(18,24)(19,28)(20,31)(21,33)(23,34)(25,39)(26,27)(35,46)(36,37)(38,40)(43,48)(45,47);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,20,22,31)(18,25,26,40)(19,29,28,41)(21,32,33,44)(23,35,36,47)(24,38,27,39)(30,43,42,48)(34,45,37,46);(10,14)(13,15)(18,26)(20,31)(23,36)(24,27)(29,41)(32,44)(34,37)(43,48)

Polycyclic presentation:

2^9|a^2=g,c^2=hi,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=ehi,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, 6970961);