#458406

order	512
nilpotency class	3
isoclinism	isoclinic class of #458406
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16316
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

726 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:

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

Polycyclic presentation:

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

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, 365602);