#8919241

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

4978 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=h,d^2=g,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8826437);