#273201

order	512
nilpotency class	3
isoclinism	isoclinic class of #273179
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3558	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

885 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180397);