#7024130

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1515 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fg,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,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, 6931326);