#9061452

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

4941 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)(11,13)(12,15)(14,16)(17,18,22,24)(19,23,20,25)(21,26,29,31)(27,30,28,32);(1,2,5,6)(3,7,8,4)(9,10,14,16)(11,13,15,12)(17,19)(18,23)(20,22)(21,27)(24,25)(26,30)(28,29)(31,32);(1,2)(3,7)(4,8)(5,6)(9,11,14,15)(10,12,16,13)(18,24)(19,20)(21,29)(30,32);(2,6)(3,8)(9,12,14,13)(10,11,16,15)(17,20,22,19)(18,25,24,23)(21,27,29,28)(26,30,31,32);(1,3)(2,4)(5,8)(6,7)(9,10)(11,12)(13,15)(14,16)(17,21,22,29)(18,26,24,31)(19,27,20,28)(23,30,25,32);(1,4)(2,3)(5,7)(6,8)(9,13,14,12)(10,15,16,11)(17,18)(19,23)(20,25)(21,26)(22,24)(27,30)(28,32)(29,31)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=g,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8968648);