#7662032

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

4392 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)(9,10,14,12)(11,15,17,18)(13,16,21,20)(19,22,23,24)(25,26,30,32)(27,35,38,31)(28,29,34,33)(36,39,40,37);(1,2,5,6)(3,7,4,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,27)(26,31)(28,36)(29,37)(30,38)(32,35)(33,39)(34,40);(1,3,5,4)(2,7,6,8)(11,17)(15,18)(19,23)(22,24)(26,32)(28,34)(31,35)(36,40);(1,3,5,4)(2,7,6,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,28,30,34)(26,33,32,29)(27,36,38,40)(31,39,35,37);(1,4,5,3)(2,8,6,7)(9,12,14,10)(11,18,17,15)(13,16,21,20)(19,22,23,24)(25,29,30,33)(26,34,32,28)(27,37,38,39)(31,40,35,36);(9,13,14,21)(10,16,12,20)(11,19,17,23)(15,22,18,24)(25,27)(26,35)(28,36)(29,39)(30,38)(31,32)(33,37)(34,40)

Polycyclic presentation:

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

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, 7569228);