#166696

order	512
nilpotency class	3
isoclinism	isoclinic class of #166289
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

773 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gi,d^2=g,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 73892);