#6461570

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1880 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)(3,6)(4,5)(7,8)(17,18,23,36,22,27,21,26)(19,28,35,24,29,43,31,38)(20,25,37,48,33,41,34,42)(30,44,47,39,32,46,45,40);(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,32)(21,35)(22,29)(23,31)(25,40)(26,43)(27,38)(28,36)(30,33)(34,45)(37,47)(39,41)(42,44)(46,48);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(19,29)(24,38)(28,43)(30,32)(31,35)(39,40)(44,46)(45,47);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,20)(18,25)(19,30)(21,34)(22,33)(23,37)(24,39)(26,42)(27,41)(28,44)(29,32)(31,45)(35,47)(36,48)(38,40)(43,46);(20,33)(25,41)(30,32)(34,37)(39,40)(42,48)(44,46)(45,47)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6368766);