#8699246

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	#2537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4587 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)(11,15)(12,14)(13,16)(17,18,24,23)(19,26,20,25)(21,28,22,27)(29,31,30,32);(1,2)(3,7)(4,8)(5,6)(9,10,12,14)(11,13,16,15)(17,19,24,20)(18,25,23,26)(21,29,22,30)(27,31,28,32);(1,3)(2,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,20,24,19)(18,26,23,25)(21,30,22,29)(27,32,28,31);(2,6)(4,7)(9,11,12,16)(10,13,14,15)(17,21,24,22)(18,27,23,28)(19,29,20,30)(25,31,26,32);(1,3)(2,7)(4,6)(5,8)(17,22,24,21)(18,28,23,27)(19,30,20,29)(25,32,26,31);(1,4,5,7)(2,8,6,3)(17,23,24,18)(19,26,20,25)(21,27,22,28)(29,31,30,32)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8606442);