#10202115

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

4063 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10109311);