#9606446

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

5003 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)(17,18,22,23)(19,27,28,24)(20,25,31,30)(21,29,32,26);(1,2,5,4)(3,7,8,6)(9,11,14,13)(10,15,12,16)(18,23)(20,31)(24,27)(26,29);(1,3,5,8)(2,6,4,7)(10,12)(11,13)(18,23)(19,28)(20,31)(21,32);(1,3,5,8)(2,7,4,6)(9,11,14,13)(10,15,12,16)(17,19)(18,24)(20,26)(21,30)(22,28)(23,27)(25,32)(29,31);(1,3)(2,7)(4,6)(5,8)(9,12,14,10)(11,15,13,16)(17,20)(18,25)(19,29)(21,24)(22,31)(23,30)(26,28)(27,32);(1,4,5,2)(3,6,8,7)(9,13,14,11)(10,15,12,16)(17,21)(18,26)(19,30)(20,24)(22,32)(23,29)(25,28)(27,31)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9513642);