#170300

order	512
nilpotency class	3
isoclinism	isoclinic class of #169613
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

681 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=ghi,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, 77496);