#6502266

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499823
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×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1892 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)(17,18,24,29)(19,25,33,42)(20,34,21,27)(22,28,23,38)(26,37,43,35)(30,40,36,44)(31,41,32,47)(39,46,48,45);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,25)(20,30)(21,36)(22,31)(23,32)(24,33)(26,39)(27,44)(28,41)(29,42)(34,40)(35,45)(37,46)(38,47)(43,48);(9,11)(10,13)(12,16)(14,15)(17,20,24,21)(18,26,29,43)(19,30,33,36)(22,35,23,37)(25,39,42,48)(27,38,34,28)(31,45,32,46)(40,41,44,47);(11,16)(13,15)(17,21,24,20)(18,27,29,34)(19,30,33,36)(22,35,23,37)(25,40,42,44)(26,38,43,28)(31,46,32,45)(39,41,48,47);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,22,24,23)(18,28,29,38)(19,31,33,32)(20,35,21,37)(25,41,42,47)(26,27,43,34)(30,45,36,46)(39,44,48,40)

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=i,e^2=fi,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,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, 6409462);