#428067

order	512
nilpotency class	3
isoclinism	isoclinic class of #428067
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15377
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2008	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1173 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)(3,11)(4,16)(5,19)(6,10)(7,15)(8,18)(9,21)(12,20)(13,27)(14,24)(17,23)(22,31)(25,28)(26,29)(30,32);(1,2)(3,12,14,7)(4,8)(5,9)(6,10)(11,15,24,20)(13,22,28,26)(16,18)(17,23)(19,21)(25,31,27,29)(30,32);(1,3)(2,7)(4,13)(5,20)(6,14)(8,22)(9,24)(10,12)(11,19)(15,21)(16,26)(17,31)(18,28)(23,27)(25,30)(29,32);(1,4,5,17,6,18,21,32)(2,8,9,23,10,16,19,30)(3,13,20,31,14,28,15,29)(7,22,24,27,12,26,11,25)

Polycyclic presentation:

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

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, 335263);