#10038468

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

5096 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gh,d^2=gi,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, 9945664);