#9184692

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

4917 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=ghi,d^2=h,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, 9091888);