#6500748

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499804
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	#35149	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2366 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)(19,24)(20,33)(21,27)(22,26)(23,28)(25,34)(29,43)(30,42)(31,38)(32,39)(35,40)(36,41)(37,45)(44,48)(46,47);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,29)(21,31)(22,30)(23,32)(25,37)(26,42)(27,38)(28,39)(33,43)(34,45)(35,46)(36,44)(40,47)(41,48);(1,3,4,7)(2,6,5,8)(9,11)(10,13)(12,16)(14,15)(17,20)(18,25)(19,29)(21,34)(22,36)(23,35)(24,37)(26,33)(27,40)(28,41)(30,44)(31,45)(32,46)(38,47)(39,48)(42,43);(11,16)(13,15)(17,21,23,22)(18,26,28,27)(19,30,32,31)(20,34,35,36)(24,38,39,42)(25,33,41,40)(29,44,46,45)(37,47,48,43);(10,14)(13,15)(17,21,23,22)(18,27,28,26)(19,31,32,30)(20,34,35,36)(24,38,39,42)(25,40,41,33)(29,45,46,44)(37,47,48,43)

Polycyclic presentation:

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

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, 6407944);