#8463240

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

4788 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=gh,d^2=gh,e^2=hi,f^2=gi,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, 8370436);