#506180

order	512
nilpotency class	4
isoclinism	isoclinic class of #506179
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#28080
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3058	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1765 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

398 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:

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

Polycyclic presentation:

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

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, 413376);