#7012308

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1375 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)(17,18,23,27,22,29,21,28)(19,24,31,46,33,42,32,41)(20,34,36,25,38,26,37,44)(30,45,48,39,35,40,47,43);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,33)(18,24,29,42)(20,35,38,30)(21,32,23,31)(25,43,44,39)(26,45,34,40)(27,46,28,41)(36,47,37,48);(9,11)(10,13)(12,16)(14,15)(17,20,22,38)(18,25,29,44)(19,30,33,35)(21,36,23,37)(24,39,42,43)(26,27,34,28)(31,47,32,48)(40,46,45,41);(3,8)(6,7)(11,16)(13,15)(17,20)(18,26)(19,30)(21,37)(22,38)(23,36)(24,40)(25,28)(27,44)(29,34)(31,48)(32,47)(33,35)(39,41)(42,45)(43,46);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(17,21,22,23)(18,27,29,28)(19,31,33,32)(20,36,38,37)(24,41,42,46)(25,34,44,26)(30,47,35,48)(39,40,43,45)

Polycyclic presentation:

2^9|a^2=fgi,b^2=fi,c^2=i,e^2=fi,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, 6919504);