#138012

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 401 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) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

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

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, 45208);