#9243813

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

4268 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gh,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=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, 9151009);