#428564

order	512
nilpotency class	3
isoclinism	isoclinic class of #428564
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15385
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

850 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 335760);