#8968183

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

4859 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)(19,20)(21,29)(22,23)(24,30)(25,26)(27,32)(28,31);(1,2,5,4)(3,7,8,6)(10,13)(11,16)(17,19)(18,20)(21,27)(22,25)(23,26)(24,28)(29,32)(30,31);(1,3,5,8)(2,6,4,7)(9,11,14,16)(10,15,13,12)(18,23)(19,25)(21,30)(28,32);(1,3,5,8)(2,7,4,6)(9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,28)(22,26)(23,25)(24,27)(29,31)(30,32);(1,3)(2,7)(4,6)(5,8)(9,13,14,10)(11,12,16,15)(17,18)(19,26)(20,25)(21,24)(22,23)(27,32)(28,31)(29,30);(1,4,5,2)(3,6,8,7)(9,12)(10,11)(13,16)(14,15)(17,21,22,30)(18,24,23,29)(19,27,25,31)(20,28,26,32)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=ghi,d^2=hi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8875379);