#167376

order	512
nilpotency class	3
isoclinism	isoclinic class of #166993
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

672 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 74572);