#424732

order	512
nilpotency class	3
isoclinism	isoclinic class of #417617
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2678
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15324
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1090 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:

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

Polycyclic presentation:

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

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, 331928);