#7010234

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×#469	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1377 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=f,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,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, 6917430);