#6380243

order	512
nilpotency class	3
isoclinism	isoclinic class of #61049
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2362 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fg,g^2=i,b^a=bf,c^a=cg,g^a=gi,d^b=df,e^b=ei,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6287439);