#7720321

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

4557 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=g,d^2=g,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7627517);