#178070

order	512
nilpotency class	3
isoclinism	isoclinic class of #177893
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

993 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 993 conjugacy classes of subgroups — too many to draw. Only its 132 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(5,8)(10,14)(11,12)(15,16)(17,18,22,26)(19,27,29,40)(20,24,31,38)(21,33,35,25)(23,34,37,30)(28,41,43,48)(32,45,44,39)(36,46,47,42);(1,2)(3,5)(4,6)(7,8)(17,19)(18,23)(20,28)(21,34)(22,29)(24,36)(25,40)(26,37)(27,33)(30,35)(31,43)(32,46)(38,47)(39,48)(41,45)(42,44);(1,3)(2,5)(4,7)(6,8)(9,10)(11,14)(12,16)(13,15)(17,20,22,31)(18,24,26,38)(19,28,29,43)(21,32,35,44)(23,36,37,47)(25,39,33,45)(27,41,40,48)(30,42,34,46);(2,6)(5,8)(9,11,13,12)(10,14,15,16)(19,29)(20,31)(23,37)(24,38)(27,40)(30,34)(32,44)(39,45)

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=i,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, 85266);