#9863337

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

4659 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)(9,10)(11,12)(13,14)(15,16)(17,18,23,20)(19,21,24,26)(22,25,31,28)(27,29,32,30);(1,2)(3,8)(4,6)(5,7)(10,14)(12,15)(19,24)(21,26)(27,32)(29,30);(1,3,5,4)(2,6,7,8)(17,19)(18,21)(20,26)(22,27)(23,24)(25,29)(28,30)(31,32);(1,2)(3,8)(4,6)(5,7)(9,11)(10,15)(12,14)(13,16)(17,20,23,18)(19,26,24,21)(22,28,31,25)(27,30,32,29);(1,2)(3,6)(4,8)(5,7)(9,12,13,15)(10,11,14,16)(17,21,23,26)(18,24,20,19)(22,30,31,29)(25,27,28,32);(1,4,5,3)(2,6,7,8)(9,11)(10,12)(13,16)(14,15)(17,22)(18,25)(19,27)(20,28)(21,29)(23,31)(24,32)(26,30)

Polycyclic presentation:

2^9|a^2=hi,c^2=h,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, 9770533);