#176168

order	512
nilpotency class	3
isoclinism	isoclinic class of #175533
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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 83364);