#138393

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	#32911	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

630 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=ghi,d^2=ghi,e^2=i,f^2=gh,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, 45589);