#470744

order	512
nilpotency class	3
isoclinism	isoclinic class of #459145
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2751
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16697
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

742 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=h,c^2=f,f^2=i,g^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,e^b=ei,f^b=fi,d^c=de,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, 377940);