#6445585

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2136 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)(5,6)(7,11)(8,9)(10,12)(13,14)(15,25)(16,22)(17,33)(18,24)(19,23)(20,28)(21,32)(26,35)(27,37)(29,42)(30,38)(31,39)(34,40)(36,43)(41,44);(2,4)(6,9)(10,11)(13,15,18,28)(14,20,24,25)(16,30,32,44)(17,29,19,27)(21,38,22,41)(23,42,33,37)(26,34,43,31)(35,39,36,40);(5,7,8,12)(6,10,9,11)(13,16)(14,21)(15,26)(17,31)(18,32)(19,34)(20,35)(22,24)(23,39)(25,36)(27,30)(28,43)(29,44)(33,40)(37,41)(38,42);(13,16)(14,22)(15,26)(17,31)(18,32)(19,34)(20,36)(21,24)(23,40)(25,35)(27,30)(28,43)(29,44)(33,39)(37,38)(41,42);(13,17,18,19)(14,23,24,33)(15,27,28,29)(16,31,32,34)(20,37,25,42)(21,39,22,40)(26,30,43,44)(35,41,36,38)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=gi,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi

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, 6352781);