#128233

order	512
nilpotency class	3
isoclinism	isoclinic class of #108504
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#376
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1745
prime radical	C₂³	#432
LCS	G ▷ C₄² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

218 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=dh,d^2=gi,e^2=h,h^2=i,b^a=bd,c^a=ch,d^a=di,f^a=fgi,e^b=eg,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35429);