#409371

order	512
nilpotency class	3
isoclinism	isoclinic class of #409371
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15018
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ 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)(3,8,13,23)(4,17)(5,12)(6,9)(7,11)(10,19)(14,28,27,31)(15,24,16,21)(18,26)(20,25)(22,30,32,29);(1,3,6,15,5,13,7,16)(2,8,9,24,12,23,11,21)(4,14,20,30,18,27,19,29)(10,22,17,28,25,32,26,31);(2,9)(3,13)(6,7)(8,21)(10,17)(11,12)(14,27)(19,20)(22,31)(23,24)(25,26)(28,32);(1,4,5,18)(2,10)(3,14,13,27)(6,20,7,19)(8,22)(9,17)(11,26)(12,25)(15,30,16,29)(21,31)(23,32)(24,28)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 316567);