#8077329

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

4478 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)(9,10,15,14)(11,18,13,16)(12,17,21,22)(19,24,20,23)(26,31)(28,32)(33,39)(36,40);(1,2)(3,7)(4,8)(5,6)(9,11,15,13)(10,16,14,18)(12,20,21,19)(17,24,22,23)(25,26)(27,28)(29,33)(30,31)(32,35)(34,36)(37,40)(38,39);(10,14)(11,13)(17,22)(19,20)(26,31)(28,32)(33,39)(36,40);(1,3,5,4)(2,7,6,8)(9,12,15,21)(10,17,14,22)(11,19,13,20)(16,23,18,24)(25,27,30,35)(26,32,31,28)(29,37,38,34)(33,36,39,40);(1,3,5,4)(2,7,6,8)(9,13,15,11)(10,18,14,16)(12,20,21,19)(17,24,22,23)(25,28,30,32)(26,27,31,35)(29,36,38,40)(33,34,39,37);(1,4,5,3)(2,8,6,7)(9,14,15,10)(11,18,13,16)(12,17,21,22)(19,23,20,24)(25,29,30,38)(26,33,31,39)(27,34,35,37)(28,36,32,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7984525);