#8592286

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

5015 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8499482);