#142281

order	512
nilpotency class	4
isoclinism	isoclinic class of #142252
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

564 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 72 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)(3,5)(4,6)(7,8)(10,15)(11,19)(12,23)(13,25)(16,31)(17,33)(18,30)(20,29)(21,37)(24,27)(26,28)(32,40)(34,39)(36,38);(5,8)(9,10,14,18)(11,20,19,16)(12,24,26,33)(13,27,25,17)(15,28,30,23)(21,38,34,40)(22,32,35,36)(29,39,31,37);(1,3)(2,5)(4,7)(6,8)(9,11,13,22,14,19,25,35)(10,16,17,32,18,20,27,36)(12,21,23,39,26,34,28,37)(15,29,24,40,30,31,33,38)

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fh,g^a=gi,c^b=cf,d^b=dg,g^b=gi,d^c=dh,e^c=ei,f^c=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, 49477);