#6500852

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499792
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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1734 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:

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

Polycyclic presentation:

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

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, 6408048);