#7917810

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

4628 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)(9,10,15,17)(11,16,20,22)(12,13,21,18)(14,19,23,24)(25,26)(27,35)(28,39)(29,34)(30,32)(31,38)(33,37)(36,40);(1,2)(3,7)(4,8)(5,6)(9,11)(10,16)(12,14)(13,19)(15,20)(17,22)(18,24)(21,23)(25,27)(26,31)(28,36)(29,37)(30,38)(32,35)(33,40)(34,39);(1,3,5,4)(2,7,6,8)(10,17)(11,20)(12,21)(19,24)(26,32)(31,35)(33,39)(34,40);(1,3,5,4)(2,7,6,8)(9,12)(10,18)(11,14)(13,17)(15,21)(16,24)(19,22)(20,23)(25,26)(27,31)(28,33)(29,34)(30,32)(35,38)(36,40)(37,39);(1,4,5,3)(2,8,6,7)(9,13,15,18)(10,12,17,21)(11,19,20,24)(14,22,23,16)(25,28,30,37)(26,33,32,39)(27,36,38,29)(31,40,35,34);(9,14)(10,19)(11,12)(13,16)(15,23)(17,24)(18,22)(20,21)(25,29,30,36)(26,34,32,40)(27,37,38,28)(31,39,35,33)

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=ghi,e^2=gi,f^2=hi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7825006);