#8346164

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

4901 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gi,d^2=gi,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8253360);