#8433742

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

5084 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,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, 8340938);