#9064892

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

5010 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,d^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=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, 8972088);