#9720362

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

4480 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)(3,8)(4,6)(5,7)(9,10,15,17)(11,20,14,16)(12,18,21,23)(13,19,24,22)(25,26,30,28)(27,31,35,32)(29,36,37,33)(34,39,40,38);(1,3,5,6)(2,4,7,8)(9,11,15,14)(10,16,17,20)(12,19,21,22)(13,23,24,18)(25,27)(26,31)(28,32)(29,34)(30,35)(33,38)(36,39)(37,40);(10,17)(12,21)(16,20)(19,22)(25,27,30,35)(26,32,28,31)(29,34,37,40)(33,39,36,38);(9,12)(10,18)(11,19)(13,16)(14,22)(15,21)(17,23)(20,24)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,38)(37,40);(1,2)(3,4)(5,7)(6,8)(9,13)(10,19)(11,18)(12,16)(14,23)(15,24)(17,22)(20,21)(25,28,30,26)(27,31,35,32)(29,33,37,36)(34,39,40,38);(1,4)(2,6)(3,7)(5,8)(9,14,15,11)(10,20,17,16)(12,22,21,19)(13,18,24,23)(25,29,30,37)(26,33,28,36)(27,34,35,40)(31,38,32,39)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9627558);