#6516440

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1390 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=fh,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,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, 6423636);