#7038698

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1390 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)(19,29)(20,34)(21,26)(22,27)(23,38)(24,32)(25,35)(28,37)(30,40)(31,44)(33,41)(36,43)(39,47)(42,46)(45,48);(1,3,5,8)(2,6,4,7)(9,10,12,13)(11,16,15,14)(17,19,22,32)(18,24,27,29)(20,31,21,30)(23,33,37,48)(25,42,43,39)(26,44,34,40)(28,41,38,45)(35,47,36,46);(9,10)(11,14)(12,13)(15,16)(17,20,22,21)(18,25,27,43)(19,30,32,31)(23,36,37,35)(24,39,29,42)(26,38,34,28)(33,47,48,46)(40,45,44,41);(3,8)(6,7)(10,13)(14,16)(17,21,22,20)(18,26,27,34)(19,31,32,30)(23,36,37,35)(24,40,29,44)(25,38,43,28)(33,47,48,46)(39,45,42,41);(1,4,5,2)(3,7,8,6)(9,11,12,15)(10,14,13,16)(18,27)(20,21)(24,29)(28,38)(30,31)(35,36)(41,45)(46,47)

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^2=i,d^2=i,e^2=fi,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=dhi,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6945894);