#9058257

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

4842 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gh,d^2=g,e^2=g,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, 8965453);