#8214531

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

5181 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:

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

Polycyclic presentation:

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

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, 8121727);