#9847627

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

3952 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)(3,11)(4,7)(5,9)(6,15)(8,13)(10,16)(12,14)(18,24)(20,29)(25,30)(27,28)(33,34,39,43)(35,40,45,37)(36,42,38,41)(44,48,46,47);(1,3,7,13)(2,8,4,11)(5,6,12,16)(9,10,14,15)(17,18)(19,25)(20,26)(21,30)(22,27)(23,24)(28,32)(29,31)(33,35)(34,40)(36,46)(37,43)(38,44)(39,45)(41,48)(42,47);(2,4)(8,11)(9,14)(10,15)(17,19,23,21)(18,25,24,30)(20,28,29,27)(22,26,32,31)(33,36,39,38)(34,41,43,42)(35,44,45,46)(37,48,40,47);(1,4,7,2)(3,11,13,8)(5,14,12,9)(6,15,16,10)(17,20)(18,26)(19,28)(21,27)(22,30)(23,29)(24,31)(25,32)(33,37,39,40)(34,35,43,45)(36,47,38,48)(41,46,42,44);(1,5)(2,9)(3,6)(4,14)(7,12)(8,10)(11,15)(13,16)(17,21,23,19)(18,25,24,30)(20,27,29,28)(22,26,32,31)(33,35)(34,40)(36,44)(37,43)(38,46)(39,45)(41,47)(42,48);(1,6)(2,10)(3,12)(4,15)(5,13)(7,16)(8,14)(9,11)(17,22,23,32)(18,27,24,28)(19,26,21,31)(20,30,29,25)(33,38,39,36)(34,42,43,41)(35,44,45,46)(37,48,40,47)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gh,d^2=h,e^2=ghi,f^2=gh,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, 9754823);