#9863344

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

4804 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gh,d^2=hi,e^2=ghi,f^2=h,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, 9770540);