#509810

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₄	D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

694 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,f^c=fh,h^c=hi,f^d=fh,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417006);