#503516

order	512
nilpotency class	4
isoclinism	isoclinic class of #501046
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	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

843 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 843 conjugacy classes of subgroups — too many to draw. Only its 182 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:

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

Polycyclic presentation:

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

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, 410712);