#9947399

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

3967 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)(3,8)(4,9)(5,10)(6,7)(11,13)(12,15)(14,16)(17,18)(19,22)(20,23)(21,26)(24,30)(25,28)(27,32)(29,31)(33,34)(35,44)(36,42)(37,43)(38,40)(39,41)(45,47)(46,48);(2,7)(3,10)(4,12)(13,14)(17,18)(19,22)(20,23)(21,24)(25,28)(26,30)(27,31)(29,32)(33,35,39,38)(34,40,41,44)(36,43,46,45)(37,42,47,48);(18,19)(23,25)(24,26)(31,32)(34,41)(36,46)(40,44)(43,45);(1,3,6,10)(2,8,7,5)(4,11,12,16)(9,13,15,14)(17,19,22,18)(20,25,28,23)(21,26,30,24)(27,32,29,31)(33,36,39,46)(34,42,41,48)(35,45,38,43)(37,44,47,40);(1,4,6,12)(2,9,7,15)(3,11,10,16)(5,14,8,13)(17,20,22,28)(18,23,19,25)(21,29,30,27)(24,32,26,31)(33,37)(34,43)(35,42)(36,44)(38,48)(39,47)(40,46)(41,45);(1,5,6,8)(2,10,7,3)(4,13,12,14)(9,11,15,16)(17,21)(18,24)(19,26)(20,27)(22,30)(23,31)(25,32)(28,29)(33,38,39,35)(34,44,41,40)(36,43,46,45)(37,42,47,48)

Polycyclic presentation:

2^9|b^2=hi,d^2=gi,e^2=gh,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, 9854595);