#7726714

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

4365 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,d^2=gh,e^2=gi,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, 7633910);