#6458524

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1984 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6365720);