#8599606

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

4875 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)(11,15,16,13)(17,18,24,28)(19,23,27,32)(20,22,31,25)(21,26,30,29);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,19,24,27)(18,23,28,32)(20,26,31,29)(21,22,30,25);(1,2,4,5)(3,6,7,8)(9,11)(10,13)(12,16)(14,15)(17,20,24,31)(18,25,28,22)(19,29,27,26)(21,23,30,32);(3,7)(6,8)(9,11)(10,13)(12,16)(14,15)(17,21)(18,26)(19,25)(20,23)(22,27)(24,30)(28,29)(31,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,22,24,25)(18,20,28,31)(19,21,27,30)(23,29,32,26);(1,3)(2,6)(4,7)(5,8)(17,23)(18,27)(19,28)(20,30)(21,31)(22,29)(24,32)(25,26)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gi,e^2=gh,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, 8506802);