#9010591

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

4797 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)(3,8)(4,7)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18,22,25)(19,23,26,30)(20,24,29,21)(27,31,32,28);(3,4)(7,8)(9,10,12,14)(11,13,16,15)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32);(1,3)(2,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(19,26)(20,29)(21,24)(23,30);(3,4)(7,8)(9,11)(10,13)(12,16)(14,15)(17,20,22,29)(18,24,25,21)(19,27,26,32)(23,31,30,28);(1,4,6,3)(2,7,5,8)(17,21,22,24)(18,20,25,29)(19,28,26,31)(23,27,30,32);(1,5,6,2)(3,7,4,8)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303333
maximal normal subgroup of	#72303333

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8917787);