#8688339

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=hi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,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, 8595535);