#504451

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	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

635 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)(13,14,20,35,34,43,33,23,18,25,17,24,19,26,37,38)(15,22,30,40,31,44,32,21,16,29,27,41,28,42,36,39);(1,2)(3,4)(13,15,18,16)(14,21,25,39)(17,32,20,36)(19,31,34,28)(22,38,29,23)(24,42,35,44)(26,41,43,40)(27,37,30,33);(2,4)(6,9)(10,12)(13,16,18,15)(14,22,25,29)(17,30,20,27)(19,31,34,28)(21,38,39,23)(24,41,35,40)(26,42,43,44)(32,33,36,37);(5,7,8,11)(6,10,9,12)(14,23)(17,33)(19,34)(20,37)(21,22)(24,26)(25,38)(27,32)(28,31)(29,39)(30,36)(35,43)(40,44)(41,42)

Polycyclic presentation:

2^9|a^2=ei,b^2=i,c^2=i,d^2=fi,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, 411647);