#409786

order	512
nilpotency class	3
isoclinism	isoclinic class of #402003
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2494
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15060
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

563 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)(4,18,22,26)(5,12)(6,10)(7,11)(8,15)(9,21,27,20)(14,29,19,23)(16,25)(17,24)(28,31,32,30);(1,3,6,15,7,17,5,16)(2,8,10,24,11,25,12,13)(4,19,21,31,22,14,20,30)(9,23,18,32,27,29,26,28);(1,4,5,20,7,22,6,21)(2,9,10,18,11,27,12,26)(3,14,15,30,17,19,16,31)(8,23,13,28,25,29,24,32);(2,10,11,12)(8,24,25,13)(9,18,27,26)(23,32,29,28)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 316982);