#7660984

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

4621 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)(11,19)(12,17)(13,18)(14,15)(16,20)(21,23)(22,24)(25,26,30,28)(27,31,33,34)(29,32,37,36)(35,38,39,40);(1,2)(3,7)(4,8)(5,6)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,3,5,4)(2,7,6,8)(10,15)(11,20)(17,24)(18,21)(27,33)(31,34)(35,39)(38,40);(9,11)(10,16)(12,21)(13,17)(14,20)(15,19)(18,22)(23,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,4,5,3)(2,8,6,7)(9,12)(10,17)(11,21)(13,16)(14,22)(15,24)(18,20)(19,23)(25,28,30,26)(27,34,33,31)(29,32,37,36)(35,38,39,40);(9,13)(10,18)(11,17)(12,19)(14,23)(15,21)(16,22)(20,24)(25,29,30,37)(26,32,28,36)(27,35,33,39)(31,38,34,40)

Polycyclic presentation:

2^9|a^2=i,c^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, 7568180);