#6439218

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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2126 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,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, 6346414);