#6492944

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1386 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)(11,15)(14,16)(17,18,23,29)(19,31,34,25)(20,36,21,27)(22,30,24,28)(26,39,45,38)(32,46,37,41)(33,42,35,43)(40,47,44,48);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,25)(20,37)(21,32)(22,33)(23,34)(24,35)(26,44)(27,41)(28,42)(29,31)(30,43)(36,46)(38,48)(39,47)(40,45);(10,13)(14,16)(17,20,23,21)(18,26,29,45)(19,32,34,37)(22,38,24,39)(25,40,31,44)(27,28,36,30)(33,47,35,48)(41,43,46,42);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,21,23,20)(18,27,29,36)(19,32,34,37)(22,38,24,39)(25,41,31,46)(26,28,45,30)(33,48,35,47)(40,43,44,42);(9,11,12,15)(10,14,13,16)(17,22,23,24)(18,28,29,30)(19,33,34,35)(20,38,21,39)(25,42,31,43)(26,36,45,27)(32,47,37,48)(40,41,44,46)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=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, 6400140);