#97716

order	512
nilpotency class	3
isoclinism	isoclinic class of #97710
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

552 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,c^b=ci,d^b=dh,e^b=egh,f^b=fgh,d^c=dgh,f^c=fg

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4912);