#507331

order	512
nilpotency class	4
isoclinism	isoclinic class of #507330
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#29051
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#25444	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1765 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

453 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)(3,13)(4,21)(5,25)(6,10)(7,12)(8,17)(9,23)(11,24)(14,30)(15,22)(16,28)(18,31)(19,29)(20,27)(26,32);(1,3,14,32,6,20,24,19,7,18,23,16,5,17,4,22)(2,8,9,29,10,13,21,28,12,27,30,15,25,31,11,26);(1,2)(3,15)(4,9)(5,10)(6,25)(7,12)(8,16)(11,14)(13,19)(17,26)(18,29)(20,28)(21,23)(22,31)(24,30)(27,32);(1,4,6,14,7,24,5,23)(2,9,25,11,12,30,10,21)(3,16,17,19,18,32,20,22)(8,26,13,29,27,28,31,15)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414527);