#93719

order	512
nilpotency class	5
isoclinism	isoclinic class of #1414
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#354
commutator	C₄²	C₂×C₁₆
Frattini, cosocle	#1831	C₂²
socle	C₂²	#1324
prime radical	C₂³	#324
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

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

normal subgroups only

This group has 126 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)(19,27,29,40)(20,31,32,47)(21,34,35,25)(22,26)(23,28,36,43)(24,33)(30,42,46,41)(37,44,38,45)(39,48);(17,19,21,29,22,30,35,46)(18,23,25,37,26,38,34,36)(20,28,33,44,32,45,48,43)(24,27,31,41,39,42,47,40)

Polycyclic presentation:

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

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, 915);