#9863273

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

4360 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=ghi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=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, 9770469);