#8600242

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

4704 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=hi,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8507438);