#425014

order	512
nilpotency class	3
isoclinism	isoclinic class of #417686
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	#15327
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1068 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)(3,13,16,30)(4,9)(5,20,22,10)(6,11)(7,12)(8,15,26,17)(14,32)(18,28)(19,29)(21,25,23,27)(24,31);(1,3)(2,8)(4,5)(6,17)(7,16)(9,27)(10,28)(11,30)(12,26)(13,31)(14,21)(15,24)(18,23)(19,22)(20,32)(25,29);(1,4,6,18,7,19,24,14)(2,9,11,28,12,29,31,32)(3,5,17,23,16,22,15,21)(8,25,30,20,26,27,13,10);(1,5)(2,10)(3,14)(4,17)(6,23)(7,22)(8,9)(11,25)(12,20)(13,32)(15,19)(16,18)(21,24)(26,29)(27,31)(28,30)

Polycyclic presentation:

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

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, 332210);