#240444

order	512
nilpotency class	3
isoclinism	isoclinic class of #233220
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

912 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:

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

Polycyclic presentation:

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

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, 147640);