#10521340

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

4155 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,d^2=hi,e^2=i,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, 10428536);