#6454775

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3521	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3066 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=i,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^c=fi,h^c=hi,f^d=fi,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, 6361971);