#138003

order	512
nilpotency class	4
isoclinism	isoclinic class of #137944
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	#2619	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#519 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 596 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=i,d^2=ghi,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,h^b=hi,d^c=di,e^c=ei,f^c=fi,f^d=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, 45199);