#430232

order	512
nilpotency class	3
isoclinism	isoclinic class of #418470
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2676
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15416
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

877 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=e,d^2=f,e^2=i,h^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,g^c=gi,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, 337428);