#7726801

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

4574 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)(9,10,14,16)(11,15,19,22)(12,17,20,23)(13,18,21,24)(25,26,32,31)(27,37,38,33)(28,34,29,35)(30,36,39,40);(1,2,5,6)(3,7,4,8)(9,11,14,19)(10,15,16,22)(12,13,20,21)(17,18,23,24)(25,27,32,38)(26,33,31,37)(28,39,29,30)(34,36,35,40);(1,3,5,4)(2,7,6,8)(10,16)(11,19)(12,20)(18,24)(25,28,32,29)(26,34,31,35)(27,30,38,39)(33,40,37,36);(1,3,5,4)(2,7,6,8)(9,12)(10,17)(11,13)(14,20)(15,18)(16,23)(19,21)(22,24)(25,29,32,28)(26,35,31,34)(27,30,38,39)(33,40,37,36);(1,3,5,4)(2,7,6,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,22)(20,23)(21,24)(25,30)(26,36)(27,28)(29,38)(31,40)(32,39)(33,35)(34,37);(1,4,5,3)(2,8,6,7)(9,13)(10,18)(11,12)(14,21)(15,17)(16,24)(19,20)(22,23)(25,31,32,26)(27,33,38,37)(28,35,29,34)(30,40,39,36)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=g,e^2=ghi,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, 7633997);