#474230

order	512
nilpotency class	3
isoclinism	isoclinic class of #474230
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16913
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

649 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381426);