#8750188

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

4909 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gi,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,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, 8657384);