#93749

order	512
nilpotency class	5
isoclinism	isoclinic class of #1418
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#357
commutator	C₁₆	C₂×C₁₆
Frattini, cosocle	C₈×C₁₆	C₂²
socle	C₂²	#1331
prime radical	C₂×D₁₆	C₈
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

101 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

Polycyclic presentation:

2^9|a^2=d,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,c^b=cei,d^b=di,e^b=eg,g^b=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226999, #72230700
maximal normal subgroup of	#72226999, #72230700
maximal quotient of	#72226997, #72230702, #72230741

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 945);