#278322

order	512
nilpotency class	3
isoclinism	isoclinic class of #278319
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3035	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

607 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 185518);