#6456053

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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1445 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)(11,14)(12,13)(15,16)(17,18,22,25)(19,23,30,38)(20,24,32,41)(21,26,35,43)(27,36,46,48)(28,37,29,39)(31,40,45,47)(33,42,34,44);(2,5)(6,8)(17,19,22,30)(18,23,25,38)(20,31,32,45)(21,28,35,29)(24,40,41,47)(26,37,43,39)(27,34,46,33)(36,44,48,42);(10,13)(14,16)(17,20,22,32)(18,24,25,41)(19,27,30,46)(21,34,35,33)(23,36,38,48)(26,44,43,42)(28,31,29,45)(37,40,39,47);(1,3,4,7)(2,6,5,8)(19,28)(20,32)(21,35)(23,37)(24,41)(26,43)(27,45)(29,30)(31,46)(36,47)(38,39)(40,48);(9,11,12,15)(10,14,13,16)(18,25)(23,38)(24,41)(26,43)(36,48)(37,39)(40,47)(42,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,d^c=di,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, 6363249);