#137984

order	512
nilpotency class	4
isoclinism	isoclinic class of #137984
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

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	#32939	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 614 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)(3,5,7,8)(4,6)(9,10)(11,21)(12,17,26,33)(13,28)(14,20)(15,31)(16,19)(18,22)(23,37,35,36)(24,30)(25,39,29,40)(27,38,32,34);(1,3,4,7)(2,5)(6,8)(9,11,22,20,15,24,30,31,14,18,21,10,16,28,13,19)(12,23,37,33,27,39,25,38,26,35,36,17,32,40,29,34);(9,12)(10,17)(11,23)(13,29)(14,26)(15,27)(16,32)(18,35)(19,34)(20,33)(21,36)(22,37)(24,39)(25,30)(28,40)(31,38)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=ghi,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, 45180);