#8132471

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

4851 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=gh,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=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, 8039667);