#9796131

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

4521 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=ghi,d^2=gh,e^2=gh,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, 9703327);