#10155803

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

5125 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)(17,18,22,23)(19,21,25,31)(20,24,29,32)(26,30,27,28);(1,2)(3,8)(4,6)(5,7)(9,10,12,14)(11,13,16,15)(17,19)(18,21)(20,27)(22,25)(23,31)(24,28)(26,29)(30,32);(1,3,5,4)(2,6,7,8)(9,10)(11,13)(12,14)(15,16)(18,23)(19,25)(24,32)(26,27);(1,2)(3,6)(4,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,19)(18,21)(20,27)(22,25)(23,31)(24,28)(26,29)(30,32);(1,4,5,3)(2,6,7,8)(17,20,22,29)(18,24,23,32)(19,26,25,27)(21,30,31,28);(1,2)(3,6)(4,8)(5,7)(17,21)(18,19)(20,28)(22,31)(23,25)(24,27)(26,32)(29,30)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=ghi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10062999);