#242689

order	512
nilpotency class	3
isoclinism	isoclinic class of #242681
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#396
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

730 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=hi,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gh,c^b=cg,g^b=gh,d^c=dh,e^c=eh,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 149885);