#179866

order	512
nilpotency class	3
isoclinism	isoclinic class of #179429
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

614 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,d^2=hi,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,d^b=dgi,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, 87062);