#8823791

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

4611 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)(6,8)(17,18)(19,27)(20,21)(22,23)(24,30)(25,26)(28,29)(31,32);(3,7)(6,8)(9,10,13,11)(12,15,16,14)(18,23)(21,25)(24,27)(29,32);(1,2)(3,6)(4,5)(7,8)(10,11)(14,15)(17,19,22,30)(18,24,23,27)(20,31,26,28)(21,29,25,32);(9,10,13,11)(12,14,16,15)(17,20,22,26)(18,25,23,21)(19,28,30,31)(24,29,27,32);(1,3,4,7)(2,6,5,8)(9,11,13,10)(12,15,16,14)(17,18)(19,27)(20,25)(21,26)(22,23)(24,30)(28,32)(29,31);(1,2)(3,6)(4,5)(7,8)(9,12,13,16)(10,14,11,15)(17,21,22,25)(18,26,23,20)(19,29,30,32)(24,28,27,31)

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=gi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8730987);