#502018

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	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

571 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)(7,9)(8,10)(11,12)(13,14)(15,25)(16,29)(17,33)(18,21)(19,36)(20,27)(22,31)(23,26)(24,34)(28,44)(30,38)(32,41)(35,39)(37,43)(40,42);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(13,15,26,42,19,28,35,31,18,16,30,43,34,32,17,27)(14,20,33,41,24,37,38,29,21,22,39,44,36,40,23,25);(2,4)(6,10)(9,12)(14,21)(15,16)(23,38)(24,36)(27,31)(28,32)(33,39)(42,43);(13,16,30,42,19,32,17,31,18,15,26,43,34,28,35,27)(14,22,39,41,24,40,23,29,21,20,33,44,36,37,38,25)

Polycyclic presentation:

2^9|b^2=efhi,d^2=ehi,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,d^c=di

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, 409214);