#9028647

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

4859 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=i,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8935843);