#7720569

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

4613 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,d^2=g,f^2=hi,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,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, 7627765);