#8458840

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

4867 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=g,e^2=ghi,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, 8366036);