#508942

order	512
nilpotency class	4
isoclinism	isoclinic class of #508298
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2250	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

803 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 803 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=i,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=di,d^c=de,f^c=fh,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, 416138);