#179987

order	512
nilpotency class	3
isoclinism	isoclinic class of #179429
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	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

670 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 87183);