#479807

order	512
nilpotency class	3
isoclinism	isoclinic class of #478580
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2789
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17256
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

512 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387003);