#8824493

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

4840 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)(10,13)(14,16)(17,18)(19,25)(20,27)(21,29)(22,24)(23,30)(26,28)(31,32);(2,5)(3,7)(11,15)(14,16)(17,18,22,24)(19,25,27,20)(21,23,30,29)(26,31,32,28);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,20)(21,26)(22,27)(23,28)(24,25)(29,31)(30,32);(2,5)(6,8)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,32)(30,31);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,20)(21,23)(22,24)(25,27)(26,28)(29,30)(31,32);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,21,22,30)(18,23,24,29)(19,26,27,32)(20,28,25,31)

Polycyclic presentation:

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

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, 8731689);