#7647233

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

4370 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)(14,22)(17,21)(19,20)(23,24)(26,29)(27,33)(32,37)(34,36);(1,2,5,6)(3,7,4,8)(9,10,15,13)(11,18,12,16)(14,17,22,21)(19,24,20,23)(25,26,30,29)(27,31,33,35)(28,32,38,37)(34,39,36,40);(1,3,5,4)(2,7,6,8)(9,11,15,12)(10,16,13,18)(14,19,22,20)(17,23,21,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,39)(36,38)(37,40);(9,12,15,11)(10,16,13,18)(14,20,22,19)(17,23,21,24)(25,27)(26,31)(28,36)(29,35)(30,33)(32,40)(34,38)(37,39);(9,13,15,10)(11,18,12,16)(14,21,22,17)(19,24,20,23)(25,28)(26,32)(27,34)(29,37)(30,38)(31,39)(33,36)(35,40);(1,4,5,3)(2,8,6,7)(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)(25,29,30,26)(27,35,33,31)(28,37,38,32)(34,40,36,39)

Polycyclic presentation:

2^9|b^2=gi,c^2=ghi,d^2=i,e^2=i,f^2=g,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, 7554429);