#8799116

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	#45248	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4644 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8706312);