#98125

order	512
nilpotency class	3
isoclinism	isoclinic class of #97300
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

507 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 507 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5321);