#8463567

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	#34681	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4781 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,d^2=g,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8370763);