#138055

order	512
nilpotency class	4
isoclinism	isoclinic class of #138028
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

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	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

582 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 84 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45251);