#142419

order	512
nilpotency class	4
isoclinism	isoclinic class of #142224
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator, Frattini, cosocle	C₂×#117	C₂³
prime radical	#3449	C₂²
LCS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

710 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 70 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=i,d^2=g,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fhi,c^b=cf,d^b=dg,h^b=hi,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49615);