#8157134

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

4424 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8064330);