#8346064

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

4992 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)(3,8)(4,6)(5,7)(10,13)(11,15)(17,18,22,25)(19,26,20,23)(21,24,29,31)(27,32,28,30);(1,2,6,4)(3,8,5,7)(9,10,12,13)(11,14,15,16)(17,19,22,20)(18,23,25,26)(21,27,29,28)(24,30,31,32);(1,2,6,4)(3,7,5,8)(9,10)(11,14)(12,13)(15,16)(21,29)(24,31)(27,28)(30,32);(1,3,6,5)(2,7,4,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,26)(21,24)(22,25)(27,30)(28,32)(29,31);(1,4,6,2)(3,7,5,8)(9,11)(10,14)(12,15)(13,16)(17,20,22,19)(18,23,25,26)(21,28,29,27)(24,30,31,32);(1,5,6,3)(2,7,4,8)(17,21,22,29)(18,24,25,31)(19,27,20,28)(23,30,26,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=hi,d^2=hi,e^2=gi,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, 8253260);