#138317

order	512
nilpotency class	4
isoclinism	isoclinic class of #138028
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	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

553 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 553 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=i,d^2=gh,e^2=i,f^2=ghi,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=fh,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, 45513);