#6489147

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469983
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1882 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)(9,10)(11,14)(12,15)(13,16)(17,18,21,26,22,25,36,42)(19,27,30,38,29,43,35,23)(20,31,34,47,33,24,39,41)(28,45,46,37,32,44,48,40);(11,12)(14,15)(17,19)(18,23)(20,32)(21,35)(22,29)(24,40)(25,38)(26,43)(27,42)(28,33)(30,36)(31,37)(34,46)(39,48)(41,44)(45,47);(1,3,5,4)(2,6,7,8)(9,11,13,12)(10,14,16,15)(17,20)(18,24)(19,28)(21,34)(22,33)(23,37)(25,31)(26,41)(27,44)(29,32)(30,46)(35,48)(36,39)(38,40)(42,47)(43,45);(9,12,13,11)(10,15,16,14)(19,29)(23,38)(27,43)(28,32)(30,35)(37,40)(44,45)(46,48);(1,4,5,3)(2,6,7,8)(10,16)(14,15)(18,25)(20,33)(23,38)(26,42)(27,43)(28,32)(34,39)(46,48)

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=hi,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,e^c=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, 6396343);