#509873

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
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

531 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)(7,8)(9,11)(10,12)(13,14,19,30,29,38,28,21,18,24,16,22,17,23,32,37)(15,20,31,41,40,44,39,33,27,36,25,34,26,35,42,43);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,18,27)(14,20,24,36)(16,25,19,31)(17,26,29,40)(21,33,37,43)(22,34,30,41)(23,35,38,44)(28,39,32,42);(2,4)(5,8,9,10)(6,7,11,12)(14,21)(16,28)(17,29)(19,32)(20,33)(22,23)(24,37)(25,39)(26,40)(30,38)(31,42)(34,35)(36,43)(41,44);(6,11)(7,12)(14,21)(16,28)(17,29)(19,32)(20,33)(22,23)(24,37)(25,39)(26,40)(30,38)(31,42)(34,35)(36,43)(41,44)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,f^c=fh,h^c=hi,f^d=fh,h^d=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, 417069);