#9682871

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

4782 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=ghi,d^2=gh,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 9590067);