#9123870

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

5213 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)(12,16)(13,14)(17,18)(19,28)(20,25)(21,22)(23,27)(24,31)(26,30)(29,32);(1,2)(3,6)(4,7)(5,8)(9,10,13,14)(11,15,16,12)(17,19)(18,24)(20,29)(21,30)(22,25)(23,31)(26,32)(27,28);(1,3)(2,5)(4,8)(6,7)(9,11)(10,12)(13,16)(14,15)(17,20,23,30)(18,25,27,26)(19,29,31,21)(22,28,32,24);(3,8)(5,6)(9,12)(10,11)(13,15)(14,16)(17,20)(18,26)(19,29)(21,31)(22,28)(23,30)(24,32)(25,27);(1,3)(2,6)(4,8)(5,7)(9,10)(11,12)(13,14)(15,16)(17,21,23,29)(18,22,27,32)(19,30,31,20)(24,25,28,26);(1,2)(3,5)(4,7)(6,8)(9,12)(10,11)(13,15)(14,16)(17,22)(18,21)(19,25)(20,28)(23,32)(24,30)(26,31)(27,29)

Polycyclic presentation:

2^9|b^2=h,c^2=gh,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265955, #72265966
maximal normal subgroup of	#72265955, #72265966
maximal quotient of	#72266508, #72266556

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9031066);