#9720544

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

5398 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gi,e^2=gh,b^a=bg,c^a=ch,d^a=di,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, 9627740);