#135455

order	512
nilpotency class	3
isoclinism	isoclinic class of #8760
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#381
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1817
prime radical	C₂⁴	C₂²⋊C₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

322 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 90 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)(3,10,12,6)(4,9)(5,8)(7,11,14,16)(13,15)(17,18,20,25,22,26,32,39,21,23,31,34,33,35,44,45)(19,27,29,40,30,41,43,48,28,24,36,37,42,38,46,47);(1,3)(2,6)(4,11)(5,12)(7,9)(8,10)(13,16)(14,15)(18,23)(20,31)(24,27)(26,35)(29,36)(32,44)(38,41)(43,46);(1,3)(2,7,8,14)(4,11)(5,12)(6,9,10,15)(13,16)(17,19,22,30,21,28,33,42)(18,24,35,41,23,27,26,38)(20,29,32,43,31,36,44,46)(25,37,45,48,34,40,39,47)

Polycyclic presentation:

2^9|a^2=f,c^2=dh,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,f^b=fg,e^c=ei,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42651);