#8387663

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5042 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=g,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8294859);