#423621

order	512
nilpotency class	3
isoclinism	isoclinic class of #423621
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15315
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1556 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)(5,20)(6,10)(7,14)(8,19)(9,22)(12,21)(13,24)(15,26)(17,23)(18,25)(27,31)(28,29)(30,32);(1,3)(2,7)(4,17)(5,21)(6,15)(8,24)(9,26)(10,12)(11,20)(13,18)(14,22)(16,29)(19,31)(23,25)(27,30)(28,32);(1,2)(3,12,15,7)(4,8)(5,9)(6,10)(11,14,26,21)(13,27,28,23)(16,19)(17,29,31,24)(18,25)(20,22)(30,32);(1,4,5,18,6,19,22,32)(2,8,9,25,10,16,20,30)(3,13,14,17,15,28,21,31)(7,23,11,24,12,27,26,29)

Polycyclic presentation:

2^9|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,d^b=de,e^b=ei,f^b=fi,h^b=hi,g^c=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, 330817);