#10247135

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

4131 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=g,d^2=ghi,e^2=gh,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, 10154331);