#6549966

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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1489 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)(3,7)(4,8)(5,6)(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);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(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);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,22,34)(18,25,26,40)(19,28,29,43)(21,32,23,33)(24,37,39,48)(27,41,35,31)(30,44,36,45)(38,47,46,42);(1,4,5,3)(2,8,6,7)(9,11)(10,14)(12,15)(13,16)(18,26)(19,29)(25,40)(27,35)(28,43)(30,36)(31,41)(44,45);(11,15)(14,16)(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,c^2=fi,d^2=f,e^2=i,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, 6457162);