#134391

order	512
nilpotency class	4
isoclinism	isoclinic class of #12826
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1804
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#418
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

382 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 116 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)(9,10,15,21,16,22,13,19)(11,23,24,35,12,26,27,17)(14,30,32,39,33,38,29,20)(18,31,36,25,34,40,37,28);(9,11,16,12)(10,17)(13,24,15,27)(14,28,33,25)(18,20)(19,23)(21,26)(22,35)(29,31,32,40)(30,37)(34,39)(36,38);(9,12,16,11)(10,18,22,34)(13,24,15,27)(14,31,33,40)(17,20,35,39)(19,37,21,36)(23,30,26,38)(25,29,28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41587);