#8587179

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

5072 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8494375);