#9912947

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

4257 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)(3,7)(4,6)(5,8)(9,10,15,17)(11,18,12,13)(14,16,23,24)(19,22,20,21)(25,26,32,29)(27,35,34,31)(28,38,30,33)(36,39,37,40);(1,2,4,6)(3,5,8,7)(9,11,15,12)(10,13,17,18)(14,19,23,20)(16,21,24,22)(25,27)(26,31)(28,36)(29,35)(30,37)(32,34)(33,39)(38,40);(1,2)(3,5)(4,6)(7,8)(25,28,32,30)(26,33,29,38)(27,36,34,37)(31,39,35,40);(1,3,4,8)(2,5,6,7)(9,12,15,11)(10,13,17,18)(14,20,23,19)(16,21,24,22)(25,29,32,26)(27,35,34,31)(28,33,30,38)(36,39,37,40);(9,13)(10,11)(12,17)(14,22)(15,18)(16,20)(19,24)(21,23)(25,30,32,28)(26,33,29,38)(27,37,34,36)(31,39,35,40);(9,14)(10,16)(11,19)(12,20)(13,21)(15,23)(17,24)(18,22)(25,31,32,35)(26,34,29,27)(28,39,30,40)(33,37,38,36)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=h,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9820143);