#97893

order	512
nilpotency class	3
isoclinism	isoclinic class of #97164
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

503 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 503 conjugacy classes of subgroups — too many to draw. Only its 86 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ghi,d^2=h,e^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=ei,f^b=fgh,d^c=di,e^c=eg,f^c=fg

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5089);