#496392

order	512
nilpotency class	3
isoclinism	isoclinic class of #382477
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2310
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#21822
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

677 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=e,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=deh,e^a=ei,c^b=ceh,f^b=fi,d^c=di,g^c=gi,g^d=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403588);