#508163

order	512
nilpotency class	4
isoclinism	isoclinic class of #508008
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33179	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

876 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 876 conjugacy classes of subgroups — too many to draw. Only its 156 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)(13,14,18,23)(15,19,24,32)(16,20,29,36)(17,30,28,22)(21,37,35,27)(25,31,38,42)(26,41,39,34)(33,44,43,40);(1,2)(3,4)(6,9)(10,12)(13,15,18,24)(14,19,23,32)(16,25,29,38)(17,26,28,39)(20,31,36,42)(21,33,35,43)(22,34,30,41)(27,40,37,44);(2,4)(7,11)(10,12)(14,20)(16,27)(17,28)(19,31)(21,22)(23,36)(25,40)(26,39)(29,37)(30,35)(32,42)(33,34)(38,44)(41,43);(5,7,8,11)(6,10,9,12)(15,24)(19,32)(25,38)(26,39)(31,42)(33,43)(34,41)(40,44)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415359);