#179206

order	512
nilpotency class	3
isoclinism	isoclinic class of #178589
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

667 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)(3,7,10,16)(4,12,14,8)(6,13,15,11)(20,27)(22,29)(23,28)(25,31)(33,34,38,42)(35,43,45,48)(36,41,37,40)(39,46,47,44);(1,3)(2,6)(4,13)(5,10)(7,12)(8,16)(9,15)(11,14)(17,18,21,24)(19,22,26,29)(20,28,27,23)(25,32,31,30);(1,3)(2,7)(4,11)(5,10)(6,12)(8,15)(9,16)(13,14)(17,19,21,26)(18,22,24,29)(20,25,27,31)(23,30,28,32)(33,35)(34,39)(36,44)(37,46)(38,45)(40,43)(41,48)(42,47);(3,10)(6,15)(7,16)(11,13)(17,20,21,27)(18,23,24,28)(19,25,26,31)(22,30,29,32)(33,36,38,37)(34,40,42,41)(35,44,45,46)(39,43,47,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,d^2=hi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eg,f^a=fi,d^b=dgh,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 86402);