#8677340

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

5746 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 24 points:

(1,2,6,5)(3,8,4,7)(11,12)(15,16);(1,3,6,4)(2,7,5,8)(9,10,14,13)(11,16,12,15)(17,18)(19,22)(20,23)(21,24);(9,11,14,12)(10,15,13,16)(17,19)(18,20)(21,23)(22,24);(2,5)(7,8)(10,13)(15,16)(17,19)(18,22)(20,24)(21,23);(1,4,6,3)(2,7,5,8)(9,12,14,11)(10,15,13,16)(17,19)(18,22)(20,24)(21,23);(1,5,6,2)(3,8,4,7)(9,13,14,10)(11,16,12,15)(17,20,21,22)(18,23,24,19)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8584536);