#504455

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	#13304	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

660 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)(13,14,16,22,30,38,43,21,17,23,31,33,18,24,29,19)(15,25,26,39,42,37,44,32,27,20,34,36,28,40,41,35);(1,2)(3,4)(14,19)(16,29)(18,30)(20,32)(21,23)(22,24)(25,35)(26,41)(28,42)(31,43)(33,38)(34,44)(36,37)(39,40);(2,4)(6,9)(10,12)(13,15,17,27)(14,20,23,25)(16,26,31,34)(18,28,30,42)(19,32,21,35)(22,36,33,39)(24,37,38,40)(29,41,43,44);(5,7,8,11)(6,10,9,12)(14,21)(16,29)(18,30)(19,23)(20,35)(22,38)(24,33)(25,32)(26,41)(28,42)(31,43)(34,44)(36,40)(37,39)

Polycyclic presentation:

2^9|a^2=e,c^2=i,d^2=fi,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,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, 411651);