#7645989

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

4522 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:

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

Polycyclic presentation:

2^9|c^2=h,d^2=g,e^2=i,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, 7553185);