#6439201

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2146 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 6346397);