#9681732

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

4645 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=hi,d^2=g,e^2=g,f^2=g,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, 9588928);