#9683105

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

4518 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)(11,16,12,15)(17,18)(19,27)(20,23)(21,25)(22,30)(24,28)(26,31)(29,32);(1,2)(3,8)(4,6)(5,7)(9,11,14,12)(10,15,13,16)(17,19,23,28)(18,24,20,27)(21,31,32,22)(25,30,29,26);(1,3,5,6)(2,4,7,8)(18,20)(24,27)(25,29)(26,30);(9,11,14,12)(10,15,13,16)(17,20,23,18)(19,27,28,24)(21,29,32,25)(22,30,31,26);(1,2)(3,4)(5,7)(6,8)(9,12,14,11)(10,15,13,16)(17,21,23,32)(18,25,20,29)(19,22,28,31)(24,26,27,30);(1,4)(2,6)(3,7)(5,8)(9,13,14,10)(11,16,12,15)(17,22)(18,26)(19,21)(20,30)(23,31)(24,25)(27,29)(28,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9590301);