#7726780

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

4626 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=g,e^2=hi,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, 7633976);