#10040758

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

4507 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=g,d^2=i,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,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, 9947954);