#8429776

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

5094 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)(17,18,23,27)(19,28,31,24)(20,22,26,32)(21,30,29,25);(1,3,5,7)(2,4,8,6)(9,11,12,15)(10,13,14,16)(17,19,23,31)(18,24,27,28)(20,29,26,21)(22,30,32,25);(1,3)(2,6)(4,8)(5,7)(9,10)(11,13)(12,14)(15,16)(17,20,23,26)(18,22,27,32)(19,29,31,21)(24,30,28,25);(1,3)(2,4)(5,7)(6,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,24)(20,22)(21,25)(23,27)(26,32)(28,31)(29,30);(9,10)(11,13)(12,14)(15,16)(17,21)(18,25)(19,20)(22,24)(23,29)(26,31)(27,30)(28,32);(1,4)(2,7)(3,8)(5,6)(17,22)(18,26)(19,30)(20,27)(21,24)(23,32)(25,31)(28,29)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,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, 8336972);