#6447710

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

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 ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3034 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)(5,6,8,9)(7,11,12,10)(13,14,18,23)(15,21,27,40)(16,28,30,20)(17,32,24,22)(19,34,36,26)(25,37,42,38)(29,35,43,44)(31,39,33,41);(1,2,3,4)(6,9)(10,11)(13,15,24,34,18,27,17,26)(14,19,32,40,23,36,22,21)(16,29,33,25,30,43,31,42)(20,37,41,35,28,38,39,44);(5,7,8,12)(6,10,9,11)(13,16,18,30)(14,20,23,28)(15,25,27,42)(17,33,24,31)(19,35,36,44)(21,38,40,37)(22,41,32,39)(26,43,34,29);(13,15,18,27)(14,21,23,40)(16,25,30,42)(17,34,24,26)(19,32,36,22)(20,38,28,37)(29,31,43,33)(35,39,44,41);(16,30)(20,28)(25,42)(29,43)(31,33)(35,44)(37,38)(39,41)

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=g,d^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,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, 6354906);