#8182023

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

5042 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8089219);