#98101

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, prime radical	C₂³	#379
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

494 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 100 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) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 48 points:

(2,6)(3,7)(4,8)(9,10,12,14,13,11,15,16)(17,18)(19,27,30,39)(20,31,33,24)(21,25)(22,26)(23,29,37,34)(28,42)(32,46,48,38)(35,40)(36,43)(41,44)(45,47);(1,2,5,3)(4,7,8,6)(9,11)(10,13)(12,14)(15,16)(17,19)(18,23)(20,28)(21,34)(22,30)(24,36)(25,39)(26,37)(27,40)(29,35)(31,45)(32,47)(33,44)(38,42)(41,46)(43,48);(1,3,5,2)(4,6,8,7)(12,15)(14,16)(17,20,22,33)(18,24,26,31)(19,28,30,44)(21,32,35,48)(23,36,37,45)(25,38,40,46)(27,41,39,42)(29,43,34,47)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ghi,d^2=h,e^2=g,f^2=gi,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, 5297);