#427669

order	512
nilpotency class	3
isoclinism	isoclinic class of #424610
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2477
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15371
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

865 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)(3,7)(4,15,17,8)(5,19)(6,10)(9,14)(11,20)(12,26,16,21)(13,22)(18,25,32,30)(23,28)(24,27,29,31);(1,3)(2,7)(4,16,17,12)(5,14)(6,13)(8,21,15,26)(9,19)(10,22)(11,23)(18,31,32,27)(20,28)(24,25,29,30);(1,4)(2,8)(3,12)(5,18)(6,17)(7,21)(9,24)(10,15)(11,25)(13,16)(14,27)(19,30)(20,32)(22,26)(23,29)(28,31);(1,3)(2,9,10,23)(4,12)(5,14)(6,13)(7,19,22,11)(8,24,15,29)(16,17)(18,27)(20,28)(21,30,26,25)(31,32)

Polycyclic presentation:

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

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, 334865);