#506202

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
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	#24266	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

722 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:

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

Polycyclic presentation:

2^9|b^2=eg,c^2=i,d^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,g^c=gh,h^c=hi

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, 413398);