#9713760

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

5380 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)(18,21)(23,25)(24,29)(30,31);(3,8)(6,7)(9,11,14,13)(10,12,16,15)(20,28)(24,29)(26,32)(30,31);(1,3)(2,6)(4,7)(5,8)(9,12)(10,13)(11,16)(14,15)(17,18,22,21)(19,25,27,23)(20,24,28,29)(26,31,32,30);(1,2,5,4)(3,7,8,6)(9,11,14,13)(10,15,16,12)(17,19,22,27)(18,23,21,25)(20,26,28,32)(24,30,29,31);(1,3)(2,7)(4,6)(5,8)(9,12)(10,11)(13,16)(14,15)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32);(1,4,5,2)(3,7,8,6)(9,13,14,11)(10,15,16,12)(17,21,22,18)(19,25,27,23)(20,29,28,24)(26,31,32,30)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,d^2=h,f^2=h,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, 9620956);