#410413

order	512
nilpotency class	3
isoclinism	isoclinic class of #396100
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2414
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15077
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

762 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:

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

Polycyclic presentation:

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

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, 317609);