#6474598

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470922
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1310 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=gi,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=dhi,e^b=ei,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6381794);