#7844557

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

4357 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=hi,d^2=h,e^2=gh,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=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, 7751753);