#174616

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:

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

Polycyclic presentation:

2^9|a^2=ghi,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, 81812);