#9101812

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

5661 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261985
maximal normal subgroup of	#72261985

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9009008);