#8612868

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

5633 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)(17,18,23,24)(19,21,20,25)(22,26,30,31)(27,29,28,32);(1,2)(3,5)(4,6)(7,8)(9,10)(11,16)(12,14)(13,15)(18,24)(19,20)(26,31)(27,28);(2,3)(6,7)(9,11,12,15)(10,13,14,16)(17,19,23,20)(18,25,24,21)(22,27,30,28)(26,32,31,29);(4,8)(6,7)(10,14)(13,16)(17,20,23,19)(18,21,24,25)(22,28,30,27)(26,29,31,32);(1,3,5,2)(4,7,8,6)(9,11)(10,13)(12,15)(14,16)(17,21)(18,19)(20,24)(22,29)(23,25)(26,27)(28,31)(30,32);(1,4)(2,6)(3,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,22,23,30)(18,26,24,31)(19,27,20,28)(21,29,25,32)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=g,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8520064);