#7649927

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

4378 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gh,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,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, 7557123);