#8741963

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

3981 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=i,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,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, 8649159);