#6551935

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1355 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)(3,8)(4,6)(5,7)(9,10,15,14)(11,12,13,16)(17,18,23,35,22,26,21,27)(19,24,36,46,29,39,30,38)(20,31,32,25,34,41,33,40)(28,42,44,37,43,47,45,48);(1,3,5,4)(2,6,7,8)(9,11,15,13)(10,16,14,12)(17,19)(18,24)(20,28)(21,30)(22,29)(23,36)(25,37)(26,39)(27,38)(31,42)(32,44)(33,45)(34,43)(35,46)(40,48)(41,47);(1,4,5,3)(2,6,7,8)(9,12,15,16)(10,11,14,13)(17,20)(18,25)(19,28)(21,32)(22,34)(23,33)(24,37)(26,40)(27,41)(29,43)(30,44)(31,35)(36,45)(38,47)(39,48)(42,46);(9,13,15,11)(10,16,14,12)(18,26)(19,29)(25,40)(27,35)(28,43)(30,36)(31,41)(44,45);(9,14,15,10)(11,12,13,16)(17,21,22,23)(18,27,26,35)(19,30,29,36)(20,32,34,33)(24,38,39,46)(25,41,40,31)(28,44,43,45)(37,47,48,42)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6459131);