#6448185

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1600 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)(3,7)(4,6)(5,8)(9,10,13,15)(11,14,12,16)(17,18)(19,24)(20,26)(21,36)(22,28)(23,27)(25,35)(29,47)(30,39)(31,42)(32,41)(33,46)(34,44)(37,40)(38,45)(43,48);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,15,16)(17,19,22,32)(18,24,28,41)(20,33,35,48)(21,31,23,37)(25,43,26,46)(27,40,36,42)(29,38,30,34)(39,44,47,45);(1,3,5,4)(2,7,8,6)(9,12,13,11)(10,14,15,16)(17,20)(18,25)(19,29)(21,34)(22,35)(23,38)(24,39)(26,28)(27,44)(30,32)(31,48)(33,37)(36,45)(40,43)(41,47)(42,46);(1,4,5,3)(2,6,8,7)(17,20)(18,26)(19,30)(21,34)(22,35)(23,38)(24,39)(25,28)(27,45)(29,32)(31,33)(36,44)(37,48)(40,43)(41,47)(42,46);(17,21,22,23)(18,27,28,36)(19,31,32,37)(20,34,35,38)(24,40,41,42)(25,44,26,45)(29,48,30,33)(39,43,47,46)

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,c^2=fgi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dhi,h^b=hi,h^c=hi,h^d=hi

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, 6355381);