#347242

order	512
nilpotency class	3
isoclinism	isoclinic class of #347226
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2232 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fi,e^d=eh,g^d=gh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254438);