#178000

order	512
nilpotency class	3
isoclinism	isoclinic class of #177893
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

717 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 717 conjugacy classes of subgroups — too many to draw. Only its 128 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) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=g,d^2=h,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fi,d^b=dgh,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 85196);