#9947361

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

4152 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=gi,e^2=h,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, 9854557);