#8701746

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

5464 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=gi,e^2=gi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8608942);