#464678

order	512
nilpotency class	3
isoclinism	isoclinic class of #453966
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1548 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eg,c^2=i,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 371874);