#6461674

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437764
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1606 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)(17,18,22,26)(19,27,32,44)(20,24,33,40)(21,35,28,25)(23,36,39,31)(29,42,30,43)(34,48,45,41)(37,47,38,46);(9,10)(11,15)(12,14)(13,16)(17,19,28,36,22,32,21,31)(18,23,35,44,26,39,25,27)(20,29,45,47,33,30,34,46)(24,37,48,43,40,38,41,42);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,24)(19,29)(21,34)(22,33)(23,37)(25,41)(26,40)(27,42)(28,45)(30,32)(31,46)(35,48)(36,47)(38,39)(43,44);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,20)(18,24)(19,30)(21,34)(22,33)(23,38)(25,41)(26,40)(27,43)(28,45)(29,32)(31,47)(35,48)(36,46)(37,39)(42,44);(20,33)(24,40)(29,30)(34,45)(37,38)(41,48)(42,43)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=h,d^2=g,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, 6368870);