#9726600

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

4932 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)(3,4)(5,8)(6,7)(9,10)(11,15)(12,13)(14,16)(17,18)(19,26)(20,24)(21,25)(22,27)(23,30)(28,32)(29,31);(1,3,5,7)(2,4,8,6)(10,13)(14,15)(17,19)(18,23)(20,28)(21,29)(22,30)(24,31)(25,32)(26,27);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,20,22,21)(18,24,27,25)(19,28,30,29)(23,31,26,32);(1,2,5,8)(3,4,7,6)(9,11,12,16)(10,14,13,15)(17,21,22,20)(18,25,27,24)(19,29,30,28)(23,32,26,31);(1,3)(2,4)(5,7)(6,8)(17,19)(18,26)(20,28)(21,29)(22,30)(23,27)(24,32)(25,31);(1,4)(2,7)(3,8)(5,6)(17,18)(19,23)(20,24)(21,25)(22,27)(26,30)(28,31)(29,32)

Polycyclic presentation:

2^9|b^2=h,c^2=ghi,d^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, 9633796);