#7726594

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

4550 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7633790);