#6492204

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470137
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×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1462 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)(11,15)(14,16)(17,18)(19,28)(20,32)(21,26)(22,27)(23,37)(24,31)(25,34)(29,45)(30,41)(33,48)(35,43)(36,40)(38,44)(39,46)(42,47);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,33)(21,36)(22,31)(23,30)(25,42)(26,41)(27,28)(29,35)(32,45)(34,46)(37,40)(38,47)(39,44)(43,48);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20,22,35)(18,25,27,44)(19,29,31,33)(21,34,23,38)(24,39,28,42)(26,43,37,32)(30,46,36,47)(40,48,41,45);(10,13)(14,16)(17,21,22,23)(18,26,27,37)(19,30,31,36)(20,34,35,38)(24,40,28,41)(25,43,44,32)(29,46,33,47)(39,48,42,45);(9,11,12,15)(10,14,13,16)(18,27)(19,31)(25,44)(26,37)(29,33)(30,36)(32,43)(46,47)

Polycyclic presentation:

2^9|a^2=fi,c^2=f,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,g^c=gi

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, 6399400);