#6452684

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1645 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)(9,10)(11,16)(12,14)(13,15)(17,18,22,27)(19,24,32,41)(20,33,35,25)(21,26,23,28)(29,45,47,38)(30,39,31,40)(34,48,46,42)(36,44,37,43);(2,5)(6,8)(17,19,22,32)(18,24,27,41)(20,34,35,46)(21,31,23,30)(25,42,33,48)(26,40,28,39)(29,37,47,36)(38,44,45,43);(9,11,13,12)(10,14,15,16)(17,20,22,35)(18,25,27,33)(19,29,32,47)(21,36,23,37)(24,38,41,45)(26,43,28,44)(30,34,31,46)(39,42,40,48);(9,12,13,11)(10,14,15,16)(19,30)(20,35)(21,23)(24,39)(25,33)(26,28)(29,46)(31,32)(34,47)(38,48)(40,41)(42,45);(1,3,4,7)(2,6,5,8)(17,21,22,23)(18,26,27,28)(19,31,32,30)(20,36,35,37)(24,40,41,39)(25,43,33,44)(29,46,47,34)(38,48,45,42)

Polycyclic presentation:

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

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, 6359880);