#472953

order	512
nilpotency class	3
isoclinism	isoclinic class of #472953
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16842
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

678 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=egh,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,e^c=ei,g^d=gi,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, 380149);