#167974

order	512
nilpotency class	3
isoclinism	isoclinic class of #167761
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

683 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

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

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 75170);