#174615

order	512
nilpotency class	3
isoclinism	isoclinic class of #174477
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

640 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:

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

Polycyclic presentation:

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

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 81811);