#8094939

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

4329 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8002135);