#441397

order	512
nilpotency class	3
isoclinism	isoclinic class of #424595
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2477
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15679
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

810 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^c=ei,e^d=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, 348593);