#6454927

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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×#3518	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3011 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=i,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dhi,e^c=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, 6362123);