#494085

order	512
nilpotency class	4
isoclinism	isoclinic class of #2218
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

690 conjugacy classes of subgroups; only the 362 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 362 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=e,d^2=i,e^2=hi,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,h^a=hi,c^b=ci

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401281);