#10247307

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4153 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=gh,d^2=ghi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

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, 10154503);