#8324358

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

4900 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,d^2=gh,e^2=g,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, 8231554);