#398434

order	512
nilpotency class	3
isoclinism	isoclinic class of #398434
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14762
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1580 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)(7,14)(8,18)(9,20)(12,25)(13,27)(15,24)(17,22)(21,32)(23,29)(26,28)(30,31);(1,3,5,14,6,15,20,25)(2,7,9,11,10,12,19,24)(4,13,17,28,18,29,31,32)(8,21,22,23,16,26,30,27);(1,2)(3,12,15,7)(4,8)(5,9)(6,10)(11,14,24,25)(13,26,29,21)(16,18)(17,22)(19,20)(23,28,27,32)(30,31);(1,4,5,17,6,18,20,31)(2,8,9,22,10,16,19,30)(3,13,14,28,15,29,25,32)(7,21,11,23,12,26,24,27)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 305630);