#8093702

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

4814 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=h,d^2=h,e^2=g,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, 8000898);