#9674988

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

4602 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)(11,15)(12,13)(14,16)(18,20)(19,27)(22,32)(25,30);(1,2,5,8)(3,4,7,6)(10,13)(14,15)(17,18,23,20)(19,24,27,29)(21,30,28,25)(22,26,32,31);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,19,23,27)(18,24,20,29)(21,22,28,32)(25,31,30,26);(9,11)(10,14)(12,16)(13,15)(17,20,23,18)(19,24,27,29)(21,30,28,25)(22,31,32,26);(1,3)(2,4)(5,7)(6,8)(17,21)(18,25)(19,22)(20,30)(23,28)(24,31)(26,29)(27,32);(1,4)(2,7)(3,8)(5,6)(17,22)(18,26)(19,28)(20,31)(21,27)(23,32)(24,25)(29,30)

Polycyclic presentation:

2^9|b^2=gi,c^2=h,d^2=h,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, 9582184);