#10146177

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

4671 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10053373);