#140558

order	512
nilpotency class	4
isoclinism	isoclinic class of #13215
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1961
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

401 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 401 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=egi,d^2=gi,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=cg,e^b=ei,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47754);