#501351

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	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

724 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)(7,10)(8,9)(11,12)(15,23)(19,34)(24,41)(25,35)(26,33)(30,44)(36,40)(39,42);(6,9)(10,12)(13,14,16,20)(15,24,23,41)(17,29,27,43)(18,32,28,22)(19,35,34,25)(21,38,37,31)(26,42,33,39)(30,40,44,36);(2,4)(5,7,8,11)(6,10,9,12)(13,15)(14,19)(16,23)(17,30)(18,33)(20,34)(21,39)(22,40)(24,29)(25,31)(26,28)(27,44)(32,36)(35,38)(37,42)(41,43);(6,9)(10,12)(15,23)(19,34)(24,41)(25,35)(26,33)(30,44)(36,40)(39,42)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=e,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,d^c=di,g^c=gh,h^c=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, 408547);