#463809

order	512
nilpotency class	3
isoclinism	isoclinic class of #455393
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2700
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16490
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

576 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 371005);