#6540627

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
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×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1502 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)(3,7)(4,6)(5,8)(17,18,23,28)(19,29,33,40)(20,34,38,43)(21,26,22,27)(24,32,41,31)(25,37,44,36)(30,39,35,42)(45,47,48,46);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,31,23,33,22,32)(18,24,27,29,28,41,26,40)(20,35,37,46,38,30,36,47)(25,42,43,48,44,39,34,45);(1,3,5,4)(2,7,8,6)(10,13)(14,16)(17,20,21,37,23,38,22,36)(18,25,27,43,28,44,26,34)(19,30,31,47,33,35,32,46)(24,39,29,45,41,42,40,48);(1,4,5,3)(2,6,8,7)(9,11)(10,14)(12,15)(13,16);(11,15)(14,16)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,36,38,37)(24,40,41,29)(25,43,44,34)(30,46,35,47)(39,45,42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=fg,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^c=ei,e^d=ehi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6447823);