#8374250

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

4853 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8281446);