#8324528

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

4682 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)(18,23)(20,26)(24,30)(28,32);(1,3,6,4)(2,5,7,8)(9,10,12,13)(11,14,15,16)(19,27)(24,30)(28,32)(29,31);(2,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,30)(28,31)(29,32);(2,7)(3,4)(9,10)(11,14)(12,13)(15,16)(17,19,22,27)(18,24,23,30)(20,28,26,32)(21,29,25,31);(1,4,6,3)(2,8,7,5)(9,11,12,15)(10,14,13,16)(17,20,22,26)(18,25,23,21)(19,28,27,32)(24,31,30,29);(1,5,6,8)(2,3,7,4)(17,21,22,25)(18,26,23,20)(19,29,27,31)(24,32,30,28)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,d^2=gh,e^2=h,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, 8231724);