#8197152

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

4453 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=g,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8104348);