#169496

order	512
nilpotency class	3
isoclinism	isoclinic class of #168909
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

714 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:

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

Polycyclic presentation:

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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 76692);