#137804

order	512
nilpotency class	4
isoclinism	isoclinic class of #137768
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

539 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 539 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ch,d^b=dgi,e^b=ei,f^b=fg,g^b=gi,d^c=di,e^c=ei,f^c=fi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45000);