#401957

order	512
nilpotency class	3
isoclinism	isoclinic class of #396612
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2850
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14826
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

944 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:

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

Polycyclic presentation:

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

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, 309153);