#6508428

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500535
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1523 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)(3,8)(4,6)(5,7)(17,18,22,26)(19,27,28,42)(20,24,32,39)(21,35,29,25)(23,36,37,31)(30,43,33,46)(34,48,44,41)(38,47,40,45);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(19,28)(20,32)(23,37)(24,39)(27,42)(31,36)(34,44)(41,48);(9,11)(10,13)(12,16)(14,15)(17,19,29,36,22,28,21,31)(18,23,35,42,26,37,25,27)(20,33,44,45,32,30,34,47)(24,40,48,43,39,38,41,46);(11,16)(13,15)(17,20,22,32)(18,24,26,39)(19,30,28,33)(21,34,29,44)(23,38,37,40)(25,41,35,48)(27,43,42,46)(31,45,36,47);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(20,32)(24,39)(30,33)(34,44)(38,40)(41,48)(43,46)(45,47)

Polycyclic presentation:

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

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, 6415624);