#499375

order	512
nilpotency class	4
isoclinism	isoclinic class of #498791
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	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

732 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 218 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)(13,14,19,30,29,38,28,21,18,24,16,22,17,23,32,37)(15,20,31,41,39,44,40,33,25,34,26,35,27,36,42,43);(1,2)(3,4)(6,9)(10,12)(13,15)(14,20)(16,26)(17,27)(18,25)(19,31)(21,33)(22,35)(23,36)(24,34)(28,40)(29,39)(30,41)(32,42)(37,43)(38,44);(2,4)(5,7,8,11)(6,10,9,12)(14,21)(16,28)(17,29)(19,32)(20,33)(22,23)(24,37)(26,40)(27,39)(30,38)(31,42)(34,43)(35,36)(41,44);(15,25)(20,34)(26,31)(27,39)(33,43)(35,41)(36,44)(40,42)

Polycyclic presentation:

2^9|a^2=fi,c^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=di,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, 406571);