#7942166

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

5178 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)(3,8)(4,6)(5,7)(9,10,13,14)(11,12,15,16)(17,18,20,22)(19,21,23,24);(1,2,6,4)(3,8,5,7)(17,18)(19,21)(20,22)(23,24);(1,3,6,5)(2,7,4,8)(10,14)(11,15);(1,2,6,4)(3,8,5,7)(9,11)(10,12)(13,15)(14,16);(1,4,6,2)(3,8,5,7)(9,12)(10,11)(13,16)(14,15)(17,19,20,23)(18,21,22,24);(1,5,6,3)(2,7,4,8)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7849362);