#9183975

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

4471 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gi,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9091171);