#8505688

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

4482 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=g,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8412884);