#6477210

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470374
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 ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1563 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=g,d^2=i,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,e^c=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, 6384406);