#7648316

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

4609 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)(10,15)(11,12)(18,22)(19,20)(26,32)(30,34)(33,36)(38,40);(1,2)(3,7)(4,8)(5,6)(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,24)(21,22)(25,26)(27,36)(28,33)(29,34)(30,35)(31,32)(37,38)(39,40);(9,11,14,12)(10,16,15,17)(13,19,21,20)(18,23,22,24)(25,27,31,28)(26,33,32,36)(29,37,35,39)(30,38,34,40);(1,3,5,4)(2,7,6,8)(9,12,14,11)(10,17,15,16)(13,19,21,20)(18,23,22,24)(25,28,31,27)(26,33,32,36)(29,37,35,39)(30,40,34,38);(1,4,5,3)(2,8,6,7)(9,13,14,21)(10,18,15,22)(11,19,12,20)(16,23,17,24)(25,29,31,35)(26,34,32,30)(27,37,28,39)(33,40,36,38);(9,10)(11,16)(12,17)(13,18)(14,15)(19,23)(20,24)(21,22)(25,30,31,34)(26,35,32,29)(27,38,28,40)(33,39,36,37)

Polycyclic presentation:

2^9|c^2=hi,d^2=g,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

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, 7555512);