#9716246

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

4027 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9623442);