#8093656

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

5193 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)(17,18,23,21)(19,26,25,22)(20,28,31,24)(27,30,29,32);(1,2)(3,6)(4,5)(7,8)(10,13)(11,15)(17,19)(18,22)(20,29)(21,26)(23,25)(24,32)(27,31)(28,30);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(18,21)(19,25)(24,28)(27,29);(9,11)(10,14)(12,15)(13,16)(17,20,23,31)(18,24,21,28)(19,27,25,29)(22,30,26,32);(1,2)(3,6)(4,5)(7,8)(17,21,23,18)(19,26,25,22)(20,28,31,24)(27,32,29,30);(1,3)(2,6)(4,7)(5,8)(17,22,23,26)(18,25,21,19)(20,30,31,32)(24,29,28,27)

Polycyclic presentation:

2^9|a^2=g,d^2=g,e^2=g,f^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, 8000852);