#138156

order	512
nilpotency class	4
isoclinism	isoclinic class of #137980
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#33190	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

445 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 445 conjugacy classes of subgroups — too many to draw. Only its 110 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,d^2=ghi,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=ei,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45352);