#242481

order	512
nilpotency class	3
isoclinism	isoclinic class of #233265
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

736 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6,8,14,10,12,9,13)(7,15,16,11,18,19,17,20)(21,22,27,33,25,24,26,29)(23,30,34,35,31,28,32,36);(1,2)(3,4)(5,7,8,16,10,18,9,17)(6,11,13,15,12,20,14,19)(21,23,25,31)(22,28,24,30)(26,32,27,34)(29,35,33,36);(2,4)(6,12)(7,16)(8,9)(11,19)(15,20)(17,18)(21,24,25,22)(23,28,31,30)(26,29,27,33)(32,36,34,35);(23,31)(28,30)(32,34)(35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 149677);