#6493441

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1428 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)(3,7)(4,8)(5,6)(11,15)(14,16)(17,18)(19,28)(20,25)(21,26)(22,24)(23,29)(27,34)(30,41)(31,35)(32,39)(33,38)(36,42)(37,43)(40,45)(44,46)(47,48);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,29)(18,23,26,28)(20,30,32,42)(22,31,34,43)(24,37,27,35)(25,36,39,41)(33,44,45,48)(38,47,40,46);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(18,24)(19,29)(22,34)(23,35)(25,38)(26,27)(28,37)(30,42)(33,45)(36,46)(39,40)(41,47);(1,4,5,3)(2,8,6,7)(10,13)(14,16)(17,20,21,32)(18,25,26,39)(19,30,29,42)(22,33,34,45)(23,36,28,41)(24,38,27,40)(31,44,43,48)(35,46,37,47);(9,11,12,15)(10,14,13,16)(18,26)(19,29)(20,32)(24,27)(31,43)(33,45)(36,41)(46,47)

Polycyclic presentation:

2^9|b^2=i,c^2=fi,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,g^c=gi,e^d=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, 6400637);