#170120

order	512
nilpotency class	3
isoclinism	isoclinic class of #169613
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

651 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)(4,9,5,8)(7,12,16,13)(18,22)(20,27)(23,31)(24,28)(25,29)(30,32)(33,34,38,42)(35,43,44,47)(36,41,37,40)(39,46,48,45);(1,3,6,14)(2,7,10,16)(4,13,5,12)(8,11,9,15)(17,18,21,24)(19,23,26,32)(20,28,27,22)(25,30,29,31)(33,35)(34,39)(36,46)(37,45)(38,44)(40,47)(41,43)(42,48);(17,19,21,26)(18,23,24,32)(20,29,27,25)(22,30,28,31)(34,40)(36,37)(39,47)(41,42)(43,48)(45,46);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,15,16,11)(18,24)(22,28)(23,32)(30,31)(35,44)(39,48)(43,47)(45,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=hi,d^2=gh,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dgh,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 77316);