#9947262

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

4028 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)(4,8)(5,6)(9,13)(10,15)(11,16)(12,14)(17,18)(19,20)(21,24)(22,30)(23,25)(26,31)(27,32)(28,29)(33,34)(35,37)(36,44)(38,46)(39,40)(41,48)(42,43)(45,47);(2,6)(3,9)(8,14)(10,11)(17,19,23,21)(18,24,25,20)(22,27,31,29)(26,32,30,28)(33,35)(34,37)(36,45)(38,43)(39,41)(40,48)(42,46)(44,47);(18,25)(20,24)(26,30)(28,32)(34,40)(35,41)(36,46)(43,47);(1,2)(3,7)(4,8)(5,6)(9,13)(10,15)(11,16)(12,14)(17,20,23,24)(18,19,25,21)(22,28,31,32)(26,27,30,29)(33,36)(34,38)(35,42)(37,47)(39,46)(40,44)(41,45)(43,48);(1,3,5,9)(2,7,6,13)(4,11,12,10)(8,16,14,15)(17,21,23,19)(18,24,25,20)(22,27,31,29)(26,28,30,32)(33,37)(34,35)(36,47)(38,42)(39,48)(40,41)(43,46)(44,45);(1,4,5,12)(2,8,6,14)(3,10,9,11)(7,15,13,16)(17,22,23,31)(18,26,25,30)(19,27,21,29)(20,28,24,32)(33,38)(34,36)(35,43)(37,45)(39,44)(40,46)(41,47)(42,48)

Polycyclic presentation:

2^9|b^2=hi,d^2=hi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,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, 9854458);