#478826

order	512
nilpotency class	3
isoclinism	isoclinic class of #478826
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#17170
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

675 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)(3,13)(4,18,22,24)(5,8)(6,11)(7,12)(9,16)(10,21,27,19)(14,23)(15,30,31,26)(17,28)(20,29,32,25);(2,8)(3,14)(4,19)(5,6)(9,13)(11,12)(16,17)(18,24)(20,32)(21,22)(25,26)(29,30);(1,3,5,17,7,16,6,14)(2,9,11,28,12,13,8,23)(4,20,19,15,22,32,21,31)(10,26,18,25,27,30,24,29);(1,4,5,19,7,22,6,21)(2,10,11,18,12,27,8,24)(3,15,14,20,16,31,17,32)(9,25,23,26,13,29,28,30)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386022);