#9794278

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

3939 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gh,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,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, 9701474);