#8908034

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

4971 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=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, 8815230);