#406919

order	512
nilpotency class	3
isoclinism	isoclinic class of #398772
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2439
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14939
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

675 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319564
maximal normal subgroup of	#72319564

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 314115);