#8402334

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

4449 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gh,d^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8309530);