#141503

order	512
nilpotency class	4
isoclinism	isoclinic class of #141503
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1972
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1752 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

344 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 76 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)(3,5,7,8)(4,6)(9,10)(11,23,26,37)(12,28,25,35)(13,20)(14,19)(15,33)(16,22)(17,24,34,30)(18,27,36,40)(21,29)(31,39)(32,38);(1,3,4,7)(2,5)(6,8)(9,11,14,12,29,40,13,30,16,26,32,25,15,27,31,24)(10,17,20,36,33,28,19,37,22,34,39,18,21,35,38,23);(3,7)(5,8)(9,12,14,27,29,30,13,11,16,25,32,40,15,24,31,26)(10,18,20,28,33,23,19,34,22,36,39,35,21,37,38,17)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=e,d^2=ghi,e^2=gh,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gi,c^b=cg,d^b=dh,e^b=ei,f^b=fh,f^c=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48699);