#7007316

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	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1442 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,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, 6914512);