#9848622

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

4444 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=ghi,d^2=g,e^2=ghi,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, 9755818);