#93884

order	512
nilpotency class	5
isoclinism	isoclinic class of #1426
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#367
commutator	C₁₆	C₂×C₁₆
Frattini, cosocle	C₈×C₁₆	C₂²
socle, prime radical	C₂²	#1372
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ C₈×C₁₆ ◁ G
LECS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

62 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

Polycyclic presentation:

2^9|a^2=d,b^2=ceg,c^2=egi,d^2=f,e^2=gi,f^2=h,g^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gi,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72230700, #72230702, #72230705
maximal normal subgroup of	#72230700, #72230702, #72230705
maximal quotient of	#72230705, #72230742

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1080);