#7865042

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

5775 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7772238);