#8324703

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

4601 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8231899);