#504375

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

560 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)(7,10)(8,9)(11,12)(13,14,19,25)(15,22,16,30)(17,26,20,37)(18,33,35,38)(21,31,40,28)(23,36,24,32)(27,39,29,43)(34,42,44,41);(1,2)(3,4)(13,15,19,16)(14,21,25,40)(17,29,20,27)(18,34,35,44)(22,38,30,33)(23,43,24,39)(26,42,37,41)(28,32,31,36);(2,4)(6,9)(10,12)(13,16,19,15)(14,22,25,30)(17,29,20,27)(18,31,35,28)(21,38,40,33)(23,41,24,42)(26,39,37,43)(32,34,36,44);(5,7,8,11)(6,10,9,12)(13,17)(14,23)(15,27)(16,29)(19,20)(21,39)(22,41)(24,25)(26,33)(30,42)(32,36)(34,44)(37,38)(40,43)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411571);