#7987303

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

4440 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)(11,14)(12,19)(13,21)(15,16)(17,22)(18,23)(20,24)(25,26,30,34)(27,28,32,31)(29,33,39,40)(35,36,37,38);(1,2,5,6)(3,7,4,8)(9,11,15,12)(10,14,16,19)(13,18,22,20)(17,24,21,23)(25,27,30,32)(26,31,34,28)(29,37,39,35)(33,36,40,38);(1,3,5,4)(2,7,6,8)(10,16)(11,12)(13,22)(23,24)(27,32)(28,31)(35,37)(36,38);(1,3,5,4)(2,7,6,8)(9,12,15,11)(10,14,16,19)(13,20,22,18)(17,24,21,23)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,37)(39,40);(1,4,5,3)(2,8,6,7)(9,13)(10,17)(11,18)(12,20)(14,24)(15,22)(16,21)(19,23)(25,28,30,31)(26,32,34,27)(29,38,39,36)(33,35,40,37);(9,14)(10,11)(12,16)(13,23)(15,19)(17,20)(18,21)(22,24)(25,29,30,39)(26,33,34,40)(27,35,32,37)(28,36,31,38)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7894499);