#240304

order	512
nilpotency class	3
isoclinism	isoclinic class of #233100
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

942 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(5,6)(7,11)(8,10)(9,12)(13,14,19,24)(15,25,29,37)(16,30,33,21)(17,22,18,23)(20,27,36,28)(26,41,43,38)(31,44,42,35)(32,40,34,39);(1,2)(3,4)(13,15,19,29)(14,20,24,36)(16,31,33,42)(17,27,18,28)(21,38,30,41)(22,25,23,37)(26,32,43,34)(35,40,44,39);(2,4)(6,10)(7,8)(11,12)(13,16,17,32,19,33,18,34)(14,21,23,40,24,30,22,39)(15,26,28,42,29,43,27,31)(20,35,25,41,36,44,37,38);(5,7,9,8)(6,11,12,10)(13,17,19,18)(14,22,24,23)(15,27,29,28)(16,32,33,34)(20,25,36,37)(21,39,30,40)(26,31,43,42)(35,41,44,38)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=e,d^2=hi,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,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, 147500);