#426703

order	512
nilpotency class	3
isoclinism	isoclinic class of #417980
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2670
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15357
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

824 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^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, 333899);