#471111

order	512
nilpotency class	3
isoclinism	isoclinic class of #459756
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2761
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16755
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

529 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:

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

Polycyclic presentation:

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

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, 378307);