#508349

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

Aliases

No aliases for this group.

Subgroups

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

540 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=i,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, 415545);