#6493776

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469995
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1501 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)(3,5)(4,6)(7,8)(11,15)(14,16)(17,18,21,26,22,25,35,41)(19,27,30,36,28,42,34,23)(20,24,33,40,32,39,47,48)(29,43,44,38,31,45,46,37);(1,3,4,7)(2,5,6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,31)(21,34)(22,28)(24,38)(25,36)(26,42)(27,41)(29,32)(30,35)(33,44)(37,39)(40,43)(45,48)(46,47);(2,6)(5,8)(10,13)(14,16)(19,28)(20,32)(23,36)(24,39)(27,42)(30,34)(33,47)(40,48);(10,13)(14,16)(17,20)(18,24)(19,29)(21,33)(22,32)(23,37)(25,39)(26,40)(27,43)(28,31)(30,44)(34,46)(35,47)(36,38)(41,48)(42,45);(9,11,12,15)(10,14,13,16)(18,25)(20,32)(23,36)(26,41)(27,42)(29,31)(33,47)(44,46)

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,d^c=di,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, 6400972);