#8936720

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

5105 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8843916);