#7172626

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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1311 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)(10,13)(11,15)(17,18,23,25,21,27,22,28)(19,29,36,38,30,24,32,40)(20,26,37,41,34,44,35,43)(31,42,46,47,33,39,45,48);(1,3,5,8)(2,6,4,7)(10,13)(11,15)(17,19)(18,24)(20,33)(21,30)(22,32)(23,36)(25,40)(26,42)(27,29)(28,38)(31,34)(35,46)(37,45)(39,44)(41,47)(43,48);(1,2,5,4)(3,6,8,7)(9,10)(11,14)(12,13)(15,16)(18,25)(19,30)(20,34)(22,23)(24,38)(26,43)(27,28)(29,40)(39,48)(41,44)(42,47)(45,46);(1,4,5,2)(3,6,8,7)(10,13)(14,16)(17,20)(18,26)(19,31)(21,34)(22,35)(23,37)(24,39)(25,41)(27,44)(28,43)(29,42)(30,33)(32,45)(36,46)(38,47)(40,48);(9,11,12,15)(10,14,13,16)(19,30)(20,34)(24,29)(26,44)(32,36)(35,37)(38,40)(41,43)

Polycyclic presentation:

2^9|a^2=fghi,b^2=fi,c^2=fi,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,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, 7079822);