#8757846

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

5232 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,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, 8665042);