#179195

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

608 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=h,d^2=hi,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, 86391);