#9127558

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
prime radical	#59029	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4843 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)(9,10,13,14)(11,15,16,12)(17,18)(19,22)(20,25)(21,26)(23,24)(27,32)(28,30)(29,31);(1,3)(2,6)(4,8)(5,7)(10,14)(12,15)(17,19,23,27)(18,22,24,32)(20,28,29,21)(25,30,31,26);(1,4)(2,7)(3,8)(5,6)(9,10)(11,12)(13,14)(15,16)(18,24)(19,27)(20,29)(26,30);(2,8)(4,6)(9,11)(10,12)(13,16)(14,15)(17,20)(18,25)(19,28)(21,27)(22,30)(23,29)(24,31)(26,32);(1,4)(2,3)(5,6)(7,8)(9,12)(10,11)(13,15)(14,16)(17,21,23,28)(18,26,24,30)(19,29,27,20)(22,31,32,25);(1,3)(2,6)(4,8)(5,7)(9,11)(10,12)(13,16)(14,15)(17,22)(18,27)(19,24)(20,30)(21,25)(23,32)(26,29)(28,31)

Polycyclic presentation:

2^9|a^2=ghi,b^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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9034754);