#274037

order	512
nilpotency class	3
isoclinism	isoclinic class of #273947
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	#24770	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

853 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=i,d^2=i,e^2=i,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=gh,e^c=ehi,f^c=fhi,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, 181233);