#8346003

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

4712 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)(3,7)(4,8)(5,6)(9,10,13,16)(11,14,19,17)(12,15,20,22)(18,21,24,23)(25,26,31,30)(27,34,29,32)(28,36,38,33)(35,39,37,40);(2,6)(3,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,21)(20,22)(23,24)(25,27,31,29)(26,32,30,34)(28,37,38,35)(33,40,36,39);(1,3)(2,4)(5,8)(6,7)(12,20)(15,22)(18,24)(21,23)(26,30)(32,34)(33,36)(39,40);(1,3)(2,4)(5,8)(6,7)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,28,31,38)(26,33,30,36)(27,35,29,37)(32,39,34,40);(1,4)(2,3)(5,7)(6,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,29,31,27)(26,34,30,32)(28,35,38,37)(33,39,36,40);(9,12)(10,15)(11,18)(13,20)(14,21)(16,22)(17,23)(19,24)(25,30,31,26)(27,34,29,32)(28,36,38,33)(35,40,37,39)

Polycyclic presentation:

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

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, 8253199);