#97699

order	512
nilpotency class	3
isoclinism	isoclinic class of #97690
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

582 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 106 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,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=eh,f^b=fgh,d^c=dh,f^c=fg

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4895);