#507168

order	512
nilpotency class	4
isoclinism	isoclinic class of #500834
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	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

619 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)(7,12)(8,10)(9,11)(13,14)(15,23)(16,22)(17,35)(18,26)(19,25)(20,27)(21,37)(24,30)(28,32)(29,39)(31,43)(33,40)(34,41)(36,38)(42,44);(1,2,3,4)(5,7,9,8)(6,10,11,12)(13,15,29,44,21,17,18,16,20,31,19,30,32,33,36,34)(14,22,38,35,28,42,25,23,27,41,26,40,37,24,39,43);(2,4)(5,8,9,7)(6,10,11,12)(13,16,18,33,32,44,29,31,20,34,36,17,21,30,19,15)(14,23,25,24,37,35,38,41,27,43,39,42,28,40,26,22);(6,11)(10,12)(13,17,19,34,21,31,36,44,20,33,29,16,32,15,18,30)(14,24,26,23,28,22,39,40,27,42,38,43,37,41,25,35)

Polycyclic presentation:

2^9|b^2=egh,c^2=e,d^2=f,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,h^a=hi,c^b=cg,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414364);