#450290

order	512
nilpotency class	3
isoclinism	isoclinic class of #417669
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2678
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15962
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

969 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)(3,15)(4,20,25,10)(5,21,27,32)(6,14)(7,12)(8,13)(9,17)(11,28,31,26)(16,24,30,23)(18,29)(19,22);(1,3,6,17,8,19,7,18)(2,9,12,15,13,29,14,22)(4,21,23,31,25,32,24,11)(5,20,26,30,27,10,28,16);(1,4,6,23,8,25,7,24)(2,10,14,30,13,20,12,16)(3,11,17,21,19,31,18,32)(5,22,28,29,27,15,26,9);(1,5)(2,11)(3,16)(4,22)(6,28)(7,26)(8,27)(9,24)(10,17)(12,32)(13,31)(14,21)(15,25)(18,20)(19,30)(23,29)

Polycyclic presentation:

2^9|a^2=efh,b^2=e,c^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,g^b=gi,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, 357486);