#9713769

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

5217 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,d^2=h,f^2=gi,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, 9620965);