#8721188

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
prime radical	#59419	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4791 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=g,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8628384);