#6435738

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432861
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 ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1419 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)(3,6)(4,5)(7,8)(11,15)(14,16)(17,18,23,25,21,27,22,28)(19,29,35,39,31,24,32,37)(20,26,36,40,33,41,34,42)(30,43,46,48,44,38,45,47);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,31)(18,24,27,29)(20,30,33,44)(22,32,23,35)(25,37,28,39)(26,38,41,43)(34,45,36,46)(40,47,42,48);(10,13)(14,16)(18,25)(22,23)(24,37)(26,40)(27,28)(29,39)(32,35)(34,36)(38,47)(41,42)(43,48)(45,46);(9,11,12,15)(10,14,13,16)(17,20)(18,26)(19,30)(21,33)(22,34)(23,36)(24,38)(25,40)(27,41)(28,42)(29,43)(31,44)(32,45)(35,46)(37,47)(39,48);(1,3,4,7)(2,6,5,8)(19,31)(20,33)(24,29)(26,41)(32,35)(34,36)(37,39)(40,42)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,c^b=ch,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, 6342934);