#6605695

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1776 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)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,36,22,27,21,26)(19,28,30,24,32,38,31,39)(20,25,33,40,35,42,34,41)(29,43,45,37,46,47,44,48);(1,3,4,7)(2,5,6,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,29)(21,30)(22,32)(23,31)(25,37)(26,38)(27,39)(28,36)(33,44)(34,45)(35,46)(40,43)(41,47)(42,48);(2,6)(5,8)(10,14)(11,15)(17,20)(18,25)(19,29)(21,34)(22,35)(23,33)(24,37)(26,41)(27,42)(28,43)(30,45)(31,44)(32,46)(36,40)(38,47)(39,48);(9,11)(10,13)(12,15)(14,16)(17,21,22,23)(18,26,27,36)(19,30,32,31)(20,33,35,34)(24,38,39,28)(25,40,42,41)(29,44,46,45)(37,43,48,47);(11,15)(13,16)(17,21,22,23)(18,26,27,36)(19,30,32,31)(20,34,35,33)(24,38,39,28)(25,41,42,40)(29,45,46,44)(37,47,48,43)

Polycyclic presentation:

2^9|a^2=fi,b^2=gh,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6512891);