#506530

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
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	C₂²×#135	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

790 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)(13,14,19,26,17,37,34,44,18,25,20,27,21,24,38,43)(15,28,31,23,16,33,35,41,30,22,32,40,29,39,36,42);(5,7,8,12)(6,9,10,11)(13,15,19,31,17,16,34,35,18,30,20,32,21,29,38,36)(14,22,26,40,37,39,44,42,25,28,27,23,24,33,43,41);(2,4)(13,16,18,29)(14,23,25,40)(15,21,30,17)(19,31,20,32)(22,26,28,27)(24,41,37,42)(33,44,39,43)(34,36,38,35);(6,10)(9,11)(13,17,18,21)(14,24,25,37)(15,29,30,16)(19,34,20,38)(22,39,28,33)(23,41,40,42)(26,43,27,44)(31,36,32,35)

Polycyclic presentation:

2^9|a^2=f,b^2=eg,c^2=i,d^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,f^c=fh,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, 413726);