#6536428

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
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	#23386	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1828 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)(19,28)(20,25)(21,26)(22,36)(23,27)(24,32)(29,41)(30,42)(31,38)(33,39)(34,40)(35,46)(37,44)(43,47)(45,48);(9,10)(11,14)(12,13)(15,16)(17,19,21,31,23,30,22,32)(18,24,36,42,27,38,26,28)(20,29,34,43,33,45,35,44)(25,37,46,48,39,47,40,41);(2,5)(6,8)(10,13)(14,16)(17,20)(18,25)(19,29)(21,34)(22,35)(23,33)(24,37)(26,40)(27,39)(28,41)(30,45)(31,43)(32,44)(36,46)(38,47)(42,48);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,15)(13,16)(19,30)(20,33)(24,38)(25,39)(28,42)(31,32)(34,35)(40,46);(11,15)(14,16)(17,21,23,22)(18,26,27,36)(19,31,30,32)(20,34,33,35)(24,28,38,42)(25,40,39,46)(29,43,45,44)(37,41,47,48)

Polycyclic presentation:

2^9|b^2=fi,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,d^c=di,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6443624);