#9189500

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

5173 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319378, #72319825, #72319870
maximal normal subgroup of	#72319378, #72319825, #72319870

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9096696);