#134688

order	512
nilpotency class	3
isoclinism	isoclinic class of #8610
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#380
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1797
prime radical	C₂⁴	C₄⋊C₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

331 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 331 conjugacy classes of subgroups — too many to draw. Only its 118 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:

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

Polycyclic presentation:

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

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, 41884);