#506509

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

803 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)(13,14,17,22,26,38,30,42,19,24,25,35,18,23,34,40)(15,20,27,36,32,43,44,39,28,37,33,21,16,29,31,41);(5,7,8,12)(6,9,10,11)(13,15,25,33,26,32,34,31,19,28,17,27,18,16,30,44)(14,20,35,21,38,43,40,41,24,37,22,36,23,29,42,39);(2,4)(13,16,19,32)(14,21,24,36)(15,26,28,18)(17,33,25,27)(20,22,37,35)(23,41,38,39)(29,42,43,40)(30,44,34,31);(6,10)(9,11)

Polycyclic presentation:

2^9|a^2=f,b^2=egi,c^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fh,g^c=gh,h^c=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413705);