#8677443

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

4854 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)(11,15)(12,14)(13,16)(17,18)(19,25)(20,30)(21,27)(22,23)(24,28)(26,29)(31,32);(1,2)(3,6)(4,7)(5,8)(9,10,12,14)(11,13,16,15)(17,19,24,22)(18,25,28,23)(20,31,29,21)(26,27,30,32);(1,3)(2,4)(5,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,26)(19,21)(22,31)(23,27)(24,29)(25,32)(28,30);(1,3)(2,6)(4,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,21,24,31)(18,27,28,32)(19,29,22,20)(23,30,25,26);(1,3)(2,6)(4,8)(5,7)(17,22,24,19)(18,25,28,23)(20,31,29,21)(26,32,30,27);(1,4,5,6)(2,7,8,3)(17,23,24,25)(18,22,28,19)(20,27,29,32)(21,26,31,30)

Polycyclic presentation:

2^9|b^2=gh,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8584639);