#97298

order	512
nilpotency class	3
isoclinism	isoclinic class of #97250
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:

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

Polycyclic presentation:

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

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, 4494);