#9848342

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

4539 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=ghi,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, 9755538);