#352340

order	512
nilpotency class	3
isoclinism	isoclinic class of #352336
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2856
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#19267	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

736 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=fh,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,c^b=ci,g^d=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, 259536);