#433923

order	512
nilpotency class	3
isoclinism	isoclinic class of #433923
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15493
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23439	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1191 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 341119);