#8810224

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

4637 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8717420);