#8421651

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

4972 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8328847);