#8436255

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

4880 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=i,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8343451);