#7660579

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

4511 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)(25,26)(27,34)(28,37)(29,35)(30,32)(31,38)(33,39)(36,40);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,24)(20,22)(25,27)(26,31)(28,36)(29,37)(30,38)(32,34)(33,40)(35,39);(1,3,5,4)(2,7,6,8)(10,14)(15,21)(16,22)(23,24)(26,32)(28,39)(31,34)(35,36);(1,4,5,3)(2,8,6,7)(9,10)(11,15)(12,16)(13,14)(17,23)(18,21)(19,24)(20,22)(25,28)(26,33)(27,36)(29,34)(30,39)(31,40)(32,37)(35,38);(9,11,13,18)(10,15,14,21)(12,19,20,17)(16,24,22,23)(25,27)(26,34)(28,36)(29,33)(30,38)(31,32)(35,39)(37,40);(9,12,13,20)(10,16,14,22)(11,17,18,19)(15,23,21,24)(25,29)(26,35)(27,37)(28,34)(30,40)(31,39)(32,36)(33,38)

Polycyclic presentation:

2^9|c^2=gh,d^2=gh,e^2=i,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, 7567775);