#9794342

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

4319 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=h,d^2=h,e^2=h,f^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, 9701538);