#172686

order	512
nilpotency class	3
isoclinism	isoclinic class of #172137
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

614 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 79882);