#8096972

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

4724 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)(3,4)(5,6)(7,8)(9,10)(11,18)(12,21)(13,20)(14,16)(15,17)(19,23)(22,24)(25,26,30,34)(27,31,36,39)(28,32,35,38)(29,33,37,40);(2,6)(3,7)(9,11,15,20)(10,13,17,18)(12,22,14,19)(16,24,21,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(1,3,5,7)(2,4,6,8)(9,12,15,14)(10,16,17,21)(11,19,20,22)(13,23,18,24)(25,27,30,36)(26,31,34,39)(28,29,35,37)(32,33,38,40);(1,4)(2,7)(3,6)(5,8)(9,13)(10,11)(12,23)(14,24)(15,18)(16,19)(17,20)(21,22)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,39)(37,40);(9,14,15,12)(10,16,17,21)(11,19,20,22)(13,24,18,23)(25,28)(26,32)(27,29)(30,35)(31,33)(34,38)(36,37)(39,40);(9,10)(11,13)(12,16)(14,21)(15,17)(18,20)(19,23)(22,24)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004168);