#8700307

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3931 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=hi,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8607503);