#9183976

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

5121 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)(10,13)(11,15)(17,18,22,26)(19,23,29,24)(20,21,25,31)(27,28,30,32);(3,7)(6,8)(9,10,14,13)(11,16,15,12)(17,19)(18,23)(20,27)(21,28)(22,29)(24,26)(25,30)(31,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,16)(14,15)(17,19,22,29)(18,23,26,24)(20,27,25,30)(21,28,31,32);(9,10,14,13)(11,12,15,16)(17,19)(18,24)(20,30)(21,28)(22,29)(23,26)(25,27)(31,32);(1,3)(2,6)(4,7)(5,8)(9,12,14,16)(10,15,13,11)(17,20)(18,21)(19,27)(22,25)(23,28)(24,32)(26,31)(29,30);(1,2)(3,6)(4,5)(7,8)(9,13,14,10)(11,16,15,12)(17,21)(18,25)(19,28)(20,26)(22,31)(23,30)(24,27)(29,32)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9091172);