#179207

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

653 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,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, 86403);