#128168

order	512
nilpotency class	4
isoclinism	isoclinic class of #7307
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#342
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×#117	C₂³
socle	C₂²	C₂×#342
prime radical	C₂⁵	C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

594 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=i,e^2=i,f^2=g,b^a=bd,d^a=dg,f^a=fhi,d^b=dh,e^b=egi,h^b=hi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35364);