#506536

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

798 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)(13,14,21,35,17,24,34,42,20,27,18,25,19,26,32,41)(15,22,36,40,16,31,44,39,30,38,29,23,28,37,33,43);(5,7,8,12)(6,9,10,11)(13,15,18,29,17,16,32,33,20,30,21,36,19,28,34,44)(14,22,25,23,24,31,41,43,27,38,35,40,26,37,42,39);(2,4)(13,16)(14,23)(15,17)(18,29)(19,30)(20,28)(21,36)(22,25)(24,39)(26,43)(27,40)(31,42)(32,44)(33,34)(35,38)(37,41);(6,10)(9,11)(13,17,20,19)(14,24,27,26)(15,28,30,16)(18,32,21,34)(22,37,38,31)(23,39,40,43)(25,41,35,42)(29,44,36,33)

Polycyclic presentation:

2^9|a^2=fi,b^2=eg,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, 413732);