#8934686

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

4925 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8841882);