#6516691

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

1361 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=i,d^2=fhi,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,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, 6423887);