#499645

order	512
nilpotency class	4
isoclinism	isoclinic class of #498795
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 760 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406841);