#6451226

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1606 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)(3,6)(4,5)(7,8)(9,10,13,15)(11,14,12,16)(17,18)(19,29)(20,33)(21,26)(22,38)(23,28)(24,27)(25,32)(30,40)(31,42)(34,44)(35,45)(36,48)(37,46)(39,43)(41,47);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19)(18,25)(20,34)(21,36)(22,31)(23,32)(24,39)(26,44)(27,42)(28,29)(30,35)(33,48)(37,47)(38,43)(40,46)(41,45);(1,3,4,7)(2,6,5,8)(10,15)(14,16)(17,20,23,21)(18,26,28,33)(19,30,32,47)(22,35,24,37)(25,40,29,41)(27,45,38,46)(31,36,39,34)(42,44,43,48);(9,12,13,11)(10,16,15,14)(17,21,23,20)(18,26,28,33)(19,30,32,47)(22,37,24,35)(25,41,29,40)(27,45,38,46)(31,36,39,34)(42,48,43,44);(17,22,23,24)(18,27,28,38)(19,31,32,39)(20,35,21,37)(25,42,29,43)(26,45,33,46)(30,36,47,34)(40,44,41,48)

Polycyclic presentation:

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

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, 6358422);