#9923905

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

3931 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)(3,11)(4,9)(5,14)(6,8)(7,13)(10,15)(12,16)(17,18)(19,22)(20,23)(21,29)(24,26)(25,32)(27,28)(30,31)(33,34)(35,39)(36,45)(37,41)(38,42)(40,44)(43,48)(46,47);(1,3,6,13)(2,7,8,11)(4,5,12,15)(9,10,16,14)(17,19,24,23)(18,22,26,20)(21,27,32,31)(25,30,29,28)(35,43)(37,46)(39,48)(41,47);(2,8)(7,11)(9,16)(10,14)(17,20,24,22)(18,19,26,23)(21,30,32,28)(25,31,29,27)(33,35,38,43)(34,39,42,48)(36,37,44,46)(40,47,45,41);(1,2)(3,7)(4,9)(5,10)(6,8)(11,13)(12,16)(14,15)(17,21,24,32)(18,25,26,29)(19,27,23,31)(20,28,22,30)(33,36,38,44)(34,40,42,45)(35,37,43,46)(39,47,48,41);(1,4)(2,9)(3,5)(6,12)(7,10)(8,16)(11,14)(13,15)(17,22,24,20)(18,19,26,23)(21,30,32,28)(25,27,29,31)(33,37)(34,41)(35,44)(36,43)(38,46)(39,40)(42,47)(45,48);(1,5)(2,10)(3,12)(4,13)(6,15)(7,16)(8,14)(9,11)(17,23,24,19)(18,22,26,20)(21,31,32,27)(25,30,29,28)(33,34)(35,39)(36,40)(37,47)(38,42)(41,46)(43,48)(44,45)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9831101);