#7798049

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

4390 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)(9,10,15,12)(11,16,18,14)(13,17,23,20)(19,24,21,22)(25,26)(27,35)(28,34)(29,32)(30,36)(31,37)(33,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,18)(10,16,12,14)(13,21,23,19)(17,22,20,24)(25,27)(26,32)(28,36)(29,31)(30,33)(34,40)(35,37)(38,39);(11,18)(14,16)(19,21)(22,24)(25,28,31,39)(26,33,37,34)(27,38,29,36)(30,32,40,35);(1,3,4,7)(2,6,5,8)(9,12,15,10)(11,14,18,16)(13,17,23,20)(19,24,21,22)(25,28)(26,34)(27,36)(29,38)(30,35)(31,39)(32,40)(33,37);(9,13,15,23)(10,17,12,20)(11,19,18,21)(14,22,16,24)(25,29,31,27)(26,35,37,32)(28,36,39,38)(30,34,40,33);(9,14)(10,11)(12,18)(13,22)(15,16)(17,19)(20,21)(23,24)(25,30,31,40)(26,36,37,38)(27,33,29,34)(28,35,39,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=h,d^2=gi,e^2=i,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,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, 7705245);