#504446

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
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

656 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)(13,14,19,30,29,38,28,20,18,24,16,22,17,23,31,34)(15,21,32,41,40,44,39,33,27,37,25,35,26,36,42,43);(1,2)(3,4)(14,20)(16,28)(17,29)(19,31)(21,33)(22,23)(24,34)(25,39)(26,40)(30,38)(32,42)(35,36)(37,43)(41,44);(2,4)(6,9)(10,12)(13,15,18,27)(14,21,24,37)(16,25,19,32)(17,26,29,40)(20,33,34,43)(22,35,30,41)(23,36,38,44)(28,39,31,42);(5,7,8,11)(6,10,9,12)(14,20)(16,28)(17,29)(19,31)(21,33)(22,23)(24,34)(25,39)(26,40)(30,38)(32,42)(35,36)(37,43)(41,44)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411642);