#6404008

order	512
nilpotency class	3
isoclinism	isoclinic class of #62307
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2039 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6311204);