#9851155

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

4464 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=ghi,d^2=gh,e^2=gi,f^2=g,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, 9758351);