#431969

order	512
nilpotency class	3
isoclinism	isoclinic class of #417980
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2670
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15450
prime radical	#23399	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1162 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=ef,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,g^d=gi,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, 339165);