#9850921

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=i,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9758117);