#9947211

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

4031 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)(3,5)(4,8)(6,7)(9,13)(10,12)(11,14)(15,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,31)(29,32)(33,34)(35,39)(36,41)(37,42)(38,40)(43,44)(45,47)(46,48);(4,11)(8,14)(10,15)(12,16)(17,18,21,24)(19,25,27,22)(20,23,29,32)(26,31,28,30)(33,35,38,42)(34,39,40,37)(36,43,45,48)(41,44,47,46);(2,7)(3,9)(8,14)(10,15)(34,40)(35,42)(41,47)(43,48);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,31)(29,32)(33,35)(34,37)(36,43)(38,42)(39,40)(41,46)(44,47)(45,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,19,21,27)(18,22,24,25)(20,28,29,26)(23,31,32,30)(33,36)(34,41)(35,43)(37,46)(38,45)(39,44)(40,47)(42,48);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,20,21,29)(18,23,24,32)(19,26,27,28)(22,30,25,31)(33,37)(34,35)(36,44)(38,39)(40,42)(41,48)(43,47)(45,46)

Polycyclic presentation:

2^9|b^2=gi,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, 9854407);