#135706

order	512
nilpotency class	3
isoclinism	isoclinic class of #8803
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#383
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1839
prime radical	C₂³	#432
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

196 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 78 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)(3,11,14,15)(4,7)(5,9)(6,10)(8,13,16,12)(17,18,21,28,23,27,37,43,22,26,36,42,20,25,34,41)(19,29,32,44,33,40,46,48,30,24,35,39,31,38,45,47);(2,7,10,9)(8,15,16,11)(17,19,20,31,22,30,23,33)(18,24,25,40,26,29,27,38)(21,35,34,46,36,32,37,45)(28,44,41,47,42,39,43,48);(1,3,4,12,6,14,5,13)(2,8,9,11,10,16,7,15)(18,25,26,27)(19,30)(24,38,29,40)(28,41,42,43)(31,33)(32,35)(39,47,44,48)(45,46)

Polycyclic presentation:

2^9|a^2=f,b^2=de,c^2=d,d^2=gi,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=ci,f^b=fi,f^c=fgi

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, 42902);