#8535349

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

5027 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8442545);