#9946952

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

4034 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,d^2=gi,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, 9854148);