#10040605

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

4699 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:

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

Polycyclic presentation:

2^9|a^2=hi,d^2=hi,e^2=g,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, 9947801);