#429675

order	512
nilpotency class	3
isoclinism	isoclinic class of #417630
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15406
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

886 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 336871);