#8824180

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

4735 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gi,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, 8731376);