#140648

order	512
nilpotency class	4
isoclinism	isoclinic class of #13229
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1964
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

446 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,c^2=egi,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dg,e^b=ei,f^b=fg,g^b=gi,d^c=di,f^c=fi,f^d=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, 47844);