#174404

order	512
nilpotency class	3
isoclinism	isoclinic class of #174093
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#46099	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

872 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 872 conjugacy classes of subgroups — too many to draw. Only its 138 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)(9,10,15,12)(11,18,22,24)(13,17,14,23)(16,20,19,21)(25,26)(27,30)(28,31)(29,32)(33,39)(34,37)(35,38)(36,40);(3,4)(6,8)(9,11,15,22)(10,16,12,19)(13,21,14,20)(17,24,23,18)(25,27)(26,30)(28,33)(29,34)(31,36)(32,37)(35,40)(38,39);(1,2,5,7)(3,6,4,8)(9,12,15,10)(11,19,22,16)(13,17,14,23)(18,21,24,20)(25,28,29,35)(26,31,32,38)(27,33,34,40)(30,36,37,39);(1,3,5,4)(2,6,7,8)(9,13,15,14)(10,17,12,23)(11,20,22,21)(16,18,19,24)(26,32)(27,34)(31,38)(33,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=gi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dh,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

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 81600);