#9848380

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

4673 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=hi,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9755576);