#6469335

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437764
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1719 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)(3,8)(4,6)(5,7)(11,15)(13,16)(17,18,22,26)(19,27,31,44)(20,24,32,40)(21,34,35,25)(23,36,39,30)(28,42,29,43)(33,48,46,41)(37,47,38,45);(9,10)(11,16)(12,14)(13,15)(17,19,21,30,22,31,35,36)(18,23,25,27,26,39,34,44)(20,28,33,45,32,29,46,47)(24,37,41,42,40,38,48,43);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,20)(18,24)(19,28)(21,33)(22,32)(23,37)(25,41)(26,40)(27,42)(29,31)(30,45)(34,48)(35,46)(36,47)(38,39)(43,44);(10,14)(13,16)(17,20)(18,24)(19,29)(21,33)(22,32)(23,38)(25,41)(26,40)(27,43)(28,31)(30,47)(34,48)(35,46)(36,45)(37,39)(42,44);(1,4,5,3)(2,6,7,8)(20,32)(24,40)(28,29)(33,46)(37,38)(41,48)(42,43)(45,47)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,e^2=gh,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6376531);