#9794321

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

4481 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)(11,20)(12,13)(14,24)(15,17)(16,21)(18,23)(19,22)(25,26)(27,28)(29,32)(30,38)(31,34)(33,39)(35,40)(36,37);(2,3)(6,7)(9,11,15,21)(10,16,17,20)(12,22,23,14)(13,24,18,19)(25,26)(27,28)(29,32)(30,33)(31,34)(35,36)(37,40)(38,39);(1,3,5,2)(4,7,8,6)(10,17)(11,21)(12,23)(19,24)(25,27,31,29)(26,32,34,28)(30,35,39,37)(33,40,38,36);(1,4,5,8)(2,6,3,7)(9,12)(10,18)(11,22)(13,17)(14,21)(15,23)(16,19)(20,24)(25,28,31,32)(26,27,34,29)(30,36,39,40)(33,35,38,37);(9,13,15,18)(10,12,17,23)(11,19,21,24)(14,20,22,16)(25,29,31,27)(26,32,34,28)(30,35,39,37)(33,36,38,40);(9,14,15,22)(10,19,17,24)(11,12,21,23)(13,20,18,16)(25,30,31,39)(26,33,34,38)(27,35,29,37)(28,36,32,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=gh,d^2=gh,e^2=h,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, 9701517);