#8656209

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

4594 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)(9,10,13,16)(11,14,18,17)(12,15,21,20)(19,22,24,23)(25,26,31,28)(27,32,29,33)(30,34,38,36)(35,39,37,40);(3,7)(6,8)(9,10)(11,17)(12,15)(13,16)(14,18)(19,23)(20,21)(22,24)(25,27)(26,32)(28,33)(29,31)(30,35)(34,39)(36,40)(37,38);(1,2,5,4)(3,6,7,8)(11,18)(14,17)(19,24)(22,23)(27,29)(32,33)(35,37)(39,40);(1,3,5,7)(2,6,4,8)(9,10)(11,14)(12,20)(13,16)(15,21)(17,18)(19,23)(22,24)(25,28,31,26)(27,33,29,32)(30,34,38,36)(35,39,37,40);(1,4,5,2)(3,8,7,6)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,23)(21,24)(25,29,31,27)(26,33,28,32)(30,37,38,35)(34,40,36,39);(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)(25,30,31,38)(26,34,28,36)(27,35,29,37)(32,39,33,40)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8563405);