#449206

order	512
nilpotency class	3
isoclinism	isoclinic class of #421157
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2610
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15936
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

594 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=f,c^2=e,d^2=eg,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,d^b=de,e^b=ei,f^c=fi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 356402);