#6463629

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437428
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1810 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gi,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,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, 6370825);