#93724

order	512
nilpotency class	4
isoclinism	isoclinic class of #1343
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#99
commutator	C₂²×C₄	C₂×C₁₆
Frattini, cosocle	C₂×C₄⋊C₁₆	C₂²
socle	C₂²	#1322
prime radical	C₂³	C₂²⋊C₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆ ◁ G
LECS	G ▷ C₂×C₄⋊C₁₆ ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

121 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 72 points:

(1,2)(3,6,8,5)(4,7)(9,10,12,18,13,19,26,39,15,20,27,40,29,41,50,62,14,17,25,35,28,36,48,57,30,37,49,58,51,59,67,70)(11,16,22,31,23,33,45,52,24,34,46,53,47,56,66,68,21,32,42,38,43,54,63,60,44,55,64,61,65,69,72,71);(2,5,7,6)(9,11,14,21)(10,16)(12,22,25,42)(13,23,28,43)(15,24,30,44)(17,32)(18,38)(19,33)(20,34)(26,45,48,63)(27,46,49,64)(29,47,51,65)(31,35)(36,54)(37,55)(39,60)(40,61)(41,56)(50,66,67,72)(52,57)(53,58)(59,69)(62,71)(68,70)

Polycyclic presentation:

2^9|a^2=d,b^2=eg,c^2=gi,d^2=f,f^2=h,h^2=i,b^a=bc,c^a=ce,e^a=eg,c^b=ci,d^b=degi,d^c=dg

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 920);