#6447598

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	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1699 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)(11,14)(12,16)(13,15)(17,18,21,25)(19,27,30,23)(20,24,32,40)(22,26,34,42)(28,45,47,37)(29,43,35,38)(31,48,46,39)(33,41,36,44);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19)(18,23)(20,31)(21,30)(22,35)(24,39)(25,27)(26,43)(28,33)(29,34)(32,46)(36,47)(37,41)(38,42)(40,48)(44,45);(9,12,13,11)(10,14,15,16)(17,20)(18,24)(19,28)(21,32)(22,36)(23,37)(25,40)(26,44)(27,45)(29,46)(30,47)(31,35)(33,34)(38,48)(39,43)(41,42);(19,29)(22,34)(23,38)(26,42)(27,43)(28,46)(30,35)(31,47)(33,36)(37,48)(39,45)(41,44);(1,3,4,7)(2,6,5,8)(19,30)(20,32)(23,27)(24,40)(29,35)(33,36)(38,43)(41,44)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,e^b=ei,h^b=hi,e^c=ei,h^c=hi,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, 6354794);