#129083

order	512
nilpotency class	3
isoclinism	isoclinic class of #108535
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄×C₄⋊C₄
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1767
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

228 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 110 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=de,d^2=gi,f^2=h,h^2=i,b^a=bd,c^a=cg,f^a=fi,c^b=ch,d^b=dg,e^b=egi,f^c=fi

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, 36279);