#98379

order	512
nilpotency class	3
isoclinism	isoclinic class of #98302
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	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

597 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 597 conjugacy classes of subgroups — too many to draw. Only its 88 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=g,d^2=h,e^2=i,f^2=h,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=ei,f^c=fhi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5575);