#8186940

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

4949 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:

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

Polycyclic presentation:

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

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, 8094136);