#9105325

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

4505 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=h,e^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,e^c=egi,e^d=egi,f^d=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, 9012521);