#178822

order	512
nilpotency class	3
isoclinism	isoclinic class of #178589
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

620 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gi,d^2=h,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dhi,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

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 86018);