#512084

order	512
nilpotency class	4
isoclinism	isoclinic class of #511741
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	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1010 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 1010 conjugacy classes of subgroups — too many to draw. Only its 186 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)(13,14,16,19)(15,24,25,42)(17,20,28,34)(18,32,29,23)(21,37,35,26)(22,39,36,30)(27,43,33,40)(31,41,44,38);(6,9)(10,12)(14,19)(20,34)(21,35)(22,36)(23,32)(24,42)(38,41)(40,43);(2,4)(5,7,8,11)(6,10,9,12)(14,20)(17,30)(18,29)(19,34)(21,24)(22,23)(26,31)(27,33)(28,39)(32,36)(35,42)(37,44)(38,43)(40,41);(13,15,16,25)(14,21,19,35)(17,31,28,44)(18,33,29,27)(20,24,34,42)(22,40,36,43)(23,41,32,38)(26,39,37,30)

Polycyclic presentation:

2^9|a^2=i,c^2=ei,d^2=i,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gh,h^a=hi,f^c=fh,g^c=gh,h^c=hi,f^d=fh,g^d=gh,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419280);