#419785

order	512
nilpotency class	3
isoclinism	isoclinic class of #418498
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2673
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15273
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1569 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,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, 326981);