#9863930

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

3933 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=hi,e^2=gi,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, 9771126);