#428806

order	512
nilpotency class	3
isoclinism	isoclinic class of #428806
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15390
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1066 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 336002);