#8230700

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

4451 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=hi,e^2=ghi,f^2=gh,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, 8137896);