#135744

order	512
nilpotency class	4
isoclinism	isoclinic class of #134690
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#106
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

248 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 52 points:

(2,4)(5,6,8,11,9,12,15,18)(7,10,14,17,13,16,19,20)(21,22)(23,31,32,43)(24,35)(25,28)(26,39)(27,33,40,45)(29,37)(30,38)(34,48,46,44)(36,49)(41,52,51,47)(42,50);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,24,33,26,34,37,47,25,32,36,45,38,46,50,52)(22,27,29,31,30,41,35,44,28,40,42,43,39,51,49,48);(1,2,3,4)(7,13)(10,16)(14,19)(17,20)(22,28)(23,32)(29,42)(30,39)(33,45)(34,46)(35,49)(47,52)

Polycyclic presentation:

2^9|a^2=f,b^2=d,c^2=ei,d^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=chi,f^b=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42940);