#6439836

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2125 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6347032);