#233022

order	512
nilpotency class	3
isoclinism	isoclinic class of #233019
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1564 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140218);