#433697

order	512
nilpotency class	3
isoclinism	isoclinic class of #420416
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15489
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

802 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)(3,9,17,29)(4,19,16,27)(5,22)(6,14)(7,12)(8,13)(10,21,30,25)(11,15)(18,28,24,32)(20,26)(23,31);(1,3,7,18,6,17,8,24)(2,9,12,28,14,29,13,32)(4,20,21,15,16,23,25,5)(10,22,19,26,30,11,27,31);(1,4,6,16)(2,10)(3,15,17,5)(7,21,8,25)(9,26)(11,28)(12,19)(13,27)(14,30)(18,23,24,20)(22,32)(29,31);(1,5,6,15)(2,11)(3,16,17,4)(7,20,8,23)(9,27)(10,28)(12,31)(13,26)(14,22)(18,25,24,21)(19,29)(30,32)

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=e,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,g^b=gi,h^b=hi,g^c=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, 340893);