#504129

order	512
nilpotency class	4
isoclinism	isoclinic class of #504129
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#26338
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3067	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

509 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411325);