#10519361

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

4137 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)(3,7)(4,8)(5,6)(9,15)(10,13)(11,16)(12,14)(17,18)(19,22)(20,29)(21,25)(23,27)(24,26)(28,32)(30,31)(33,34)(35,42)(36,45)(37,40)(38,44)(39,41)(43,47)(46,48);(2,6)(7,13)(8,14)(15,16)(17,19)(18,24)(20,28)(21,31)(22,27)(23,26)(25,32)(29,30)(33,35)(34,38)(36,46)(37,43)(39,44)(40,48)(41,42)(45,47);(17,20,23,21)(18,25,27,29)(19,28,26,31)(22,30,24,32)(33,36,39,37)(34,40,41,45)(35,43,44,46)(38,47,42,48);(1,2)(3,7)(4,8)(5,6)(9,15)(10,13)(11,16)(12,14)(17,21,23,20)(18,25,27,29)(19,28,26,31)(22,32,24,30)(33,35)(34,38)(36,43)(37,46)(39,44)(40,47)(41,42)(45,48);(1,3,5,10)(2,7,6,13)(4,11,12,9)(8,16,14,15)(17,19)(18,22)(20,28)(21,31)(23,26)(24,27)(25,30)(29,32)(33,37,39,36)(34,40,41,45)(35,43,44,46)(38,48,42,47);(1,4,5,12)(2,8,6,14)(3,9,10,11)(7,15,13,16)(17,22)(18,26)(19,27)(20,30)(21,32)(23,24)(25,31)(28,29)(33,38)(34,35)(36,47)(37,48)(39,42)(40,43)(41,44)(45,46)

Polycyclic presentation:

2^9|c^2=h,d^2=hi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10426557);