#6454914

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×#3521	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3060 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)(13,14,18,23)(15,25,28,37)(16,30,32,21)(17,24,19,22)(20,29,36,27)(26,41,42,38)(31,44,43,35)(33,39,34,40);(1,2,3,4)(13,15,19,29,18,28,17,27)(14,20,24,37,23,36,22,25)(16,31,34,42,32,43,33,26)(21,38,40,44,30,41,39,35);(6,9)(10,12)(13,16)(14,21)(15,26)(17,34)(18,32)(19,33)(20,35)(22,40)(23,30)(24,39)(25,38)(27,31)(28,42)(29,43)(36,44)(37,41);(15,27)(17,19)(20,25)(22,24)(26,31)(28,29)(33,34)(35,38)(36,37)(39,40)(41,44)(42,43);(5,7,8,11)(6,10,9,12)(16,32)(21,30)(26,42)(31,43)(33,34)(35,44)(38,41)(39,40)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,e^2=gi,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, 6362110);