#93765

order	512
nilpotency class	5
isoclinism	isoclinic class of #7418
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#357
commutator	C₂×C₁₆	C₂×C₈
Frattini, cosocle	C₈×C₁₆	C₂²
socle, prime radical	C₂²	#1331
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ C₈×C₁₆ ◁ G
LECS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

86 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=i,c^2=eghi,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=cehi,d^b=dhi,e^b=eg,g^b=gi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 961);