#10036713

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

4977 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)(3,7)(4,5)(6,8)(9,10,14,12)(11,16,13,15)(17,18)(19,24)(20,30)(21,27)(22,31)(23,25)(26,29)(28,32);(2,5)(6,7)(9,11,14,13)(10,15,12,16)(17,19,23,28)(18,24,25,32)(20,22,29,27)(21,30,31,26);(2,5)(6,7)(9,11,14,13)(10,16,12,15)(18,25)(19,28)(20,29)(21,31);(1,2)(3,6)(4,5)(7,8)(9,10,14,12)(11,16,13,15)(17,20,23,29)(18,26,25,30)(19,27,28,22)(21,24,31,32);(1,2)(3,7)(4,5)(6,8)(9,12,14,10)(11,16,13,15)(17,21,23,31)(18,27,25,22)(19,26,28,30)(20,24,29,32);(1,3)(2,6)(4,8)(5,7)(9,13,14,11)(10,15,12,16)(17,22,23,27)(18,21,25,31)(19,29,28,20)(24,30,32,26)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=ghi,d^2=gh,e^2=gh,f^2=gh,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, 9943909);