#137986

order	512
nilpotency class	4
isoclinism	isoclinic class of #137984
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 577 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45182);