#9545632

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

4311 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=gh,d^2=hi,e^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi

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, 9452828);