#6520874

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1410 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)(3,6)(4,5)(7,8)(17,18,21,23)(19,27,29,42)(20,25,32,37)(22,35,34,26)(24,38,36,30)(28,43,31,46)(33,48,44,41)(39,47,40,45);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(18,23)(24,36)(25,37)(26,35)(27,42)(39,40)(41,48)(43,46);(10,13)(11,15)(17,19,22,30,21,29,34,38)(18,24,26,27,23,36,35,42)(20,31,33,47,32,28,44,45)(25,40,41,46,37,39,48,43);(9,11,12,15)(10,14,13,16)(17,20)(18,25)(19,28)(21,32)(22,33)(23,37)(24,39)(26,41)(27,43)(29,31)(30,45)(34,44)(35,48)(36,40)(38,47)(42,46);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(20,32)(25,37)(28,31)(33,44)(39,40)(41,48)(43,46)(45,47)

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dhi,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, 6428070);