#9125926

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

4818 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=g,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9033122);