#8505781

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

4710 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)(6,7)(9,10)(11,17)(12,16)(13,22)(14,19)(15,20)(18,23)(21,24)(25,26,31,29)(27,34,32,28)(30,33,39,37)(35,38,40,36);(1,2,5,4)(3,7,8,6)(9,11,15,12)(10,16,20,17)(13,21,23,19)(14,18,24,22)(26,29)(28,34)(33,37)(36,38);(1,3,5,8)(2,6,4,7)(9,11,15,12)(10,17,20,16)(13,19,23,21)(14,18,24,22)(25,27)(26,28)(29,34)(30,35)(31,32)(33,36)(37,38)(39,40);(1,4,5,2)(3,7,8,6)(9,12,15,11)(10,17,20,16)(13,21,23,19)(14,18,24,22)(25,28,31,34)(26,32,29,27)(30,38,39,36)(33,35,37,40);(9,13,15,23)(10,18,20,22)(11,19,12,21)(14,17,24,16)(25,29,31,26)(27,34,32,28)(30,37,39,33)(35,38,40,36);(9,14,15,24)(10,19,20,21)(11,18,12,22)(13,17,23,16)(25,30,31,39)(26,33,29,37)(27,35,32,40)(28,36,34,38)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,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, 8412977);