#6474094

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470894
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	C₂×#466	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1356 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)(9,10)(11,13)(12,14)(15,16)(17,18)(19,28)(20,25)(21,26)(22,23)(24,32)(27,37)(29,45)(30,38)(31,43)(33,48)(34,42)(35,39)(36,44)(40,46)(41,47);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(18,23)(24,38)(25,39)(26,27)(28,43)(40,41)(42,44)(45,48);(1,3,5,4)(2,7,8,6)(11,15)(13,16)(17,19,21,32,22,31,37,30)(18,24,27,28,23,38,26,43)(20,33,36,47,35,29,34,46)(25,41,42,48,39,40,44,45);(1,4,5,3)(2,6,8,7)(11,15)(13,16)(17,20,22,35)(18,25,23,39)(19,29,31,33)(21,36,37,34)(24,40,38,41)(26,44,27,42)(28,45,43,48)(30,47,32,46);(10,14)(13,16)(17,21,22,37)(18,26,23,27)(19,30,31,32)(20,34,35,36)(24,28,38,43)(25,42,39,44)(29,46,33,47)(40,48,41,45)

Polycyclic presentation:

2^9|b^2=fi,c^2=fgi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,d^c=di,e^c=ei,e^d=ei

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, 6381290);