#8345962

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

4450 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)(9,10,15,14)(11,18,13,16)(12,20,22,17)(19,23,21,24)(25,26)(27,28)(29,37)(30,31)(32,39)(33,36)(34,38)(35,40);(2,6)(3,8)(9,11,15,13)(10,16,14,18)(12,21,22,19)(17,24,20,23)(25,27,30,33)(26,28,31,36)(29,34,39,40)(32,35,37,38);(1,3)(2,4)(5,8)(6,7)(10,14)(16,18)(17,20)(23,24)(26,31)(27,33)(32,37)(34,40);(1,3)(2,4)(5,8)(6,7)(9,12,15,22)(10,17,14,20)(11,19,13,21)(16,23,18,24)(25,28)(26,27)(29,38)(30,36)(31,33)(32,40)(34,37)(35,39);(1,4)(2,3)(5,7)(6,8)(9,13,15,11)(10,18,14,16)(12,19,22,21)(17,23,20,24)(25,29,30,39)(26,32,31,37)(27,34,33,40)(28,35,36,38);(9,14,15,10)(11,18,13,16)(12,20,22,17)(19,24,21,23)(25,26)(27,28)(29,32)(30,31)(33,36)(34,35)(37,39)(38,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,d^2=hi,e^2=gi,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, 8253158);