#8522150

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

5314 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:

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

Polycyclic presentation:

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

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, 8429346);