#299578

order	512
nilpotency class	3
isoclinism	isoclinic class of #299541
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#25741	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1012 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:

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

Polycyclic presentation:

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

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, 206774);