#8657249

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

5502 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8564445);