#93690

order	512
nilpotency class	5
isoclinism	isoclinic class of #1406
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#353
commutator	C₄²	C₂×C₁₆
Frattini, cosocle	#1831	C₂²
socle	C₂²	#1318
prime radical	C₂³	#323
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1831 ◁ G
LECS	G ▷ #1831 ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁

163 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 163 conjugacy classes of subgroups — too many to draw. Only its 36 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 — 2 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=i,c^2=g,d^2=f,e^2=i,f^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gi,c^b=cg,d^b=degi,e^b=ei,d^c=dgi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 886);