#8232434

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

4412 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)(11,12)(13,21)(14,15)(16,23)(17,20)(18,24)(19,22)(25,26,32,31)(27,36,30,33)(28,35,29,34)(37,39,38,40);(1,2)(3,4)(5,6)(7,8)(9,11,14,17)(10,12,15,20)(13,22,23,18)(16,24,21,19)(25,27,32,30)(26,33,31,36)(28,38,29,37)(34,40,35,39);(2,6)(4,7)(10,15)(11,17)(16,21)(18,22)(25,28,32,29)(26,34,31,35)(27,37,30,38)(33,39,36,40);(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,22)(23,24)(25,29,32,28)(26,35,31,34)(27,38,30,37)(33,40,36,39);(1,3)(2,7)(4,6)(5,8)(9,13)(10,16)(11,18)(12,19)(14,23)(15,21)(17,22)(20,24)(25,30,32,27)(26,36,31,33)(28,38,29,37)(34,40,35,39);(1,4,5,7)(2,3,6,8)(9,10)(11,12)(13,16)(14,15)(17,20)(18,19)(21,23)(22,24)(25,31,32,26)(27,36,30,33)(28,35,29,34)(37,40,38,39)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8139630);