#479860

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	#17288
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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gh,c^2=f,d^2=e,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,e^b=ei,f^b=fi,d^c=de,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, 387056);