#278362

order	512
nilpotency class	3
isoclinism	isoclinic class of #278359
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

591 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 185558);