#402455

order	512
nilpotency class	3
isoclinism	isoclinic class of #393893
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1436 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,5,7)(3,8,4,6)(9,10,12,14)(11,13,15,16)(17,18,21,26,22,25,31,32)(19,27,28,24,20,29,30,23);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,20)(18,23,25,24)(21,30,31,28)(26,29,32,27);(1,4,5,3)(2,6,7,8)(11,15)(13,16)(17,20,22,19)(18,24,25,23)(21,28,31,30)(26,27,32,29);(18,25)(21,31)(23,24)(28,30)

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,c^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264431, #72264501, #72264524
maximal normal subgroup of	#72264431, #72264501, #72264524

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 309651);