#8721202

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	#35715	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5016 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^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, 8628398);