#423245

order	512
nilpotency class	3
isoclinism	isoclinic class of #423245
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15313
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60644	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1516 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)(3,14)(4,9)(5,10)(6,13)(7,12)(8,16)(11,18)(15,26)(17,32)(19,30)(20,29)(21,27)(22,28)(23,25)(24,31);(1,3,6,16,7,5,18,23)(2,8,11,14,12,25,13,10)(4,17,20,15,22,31,21,19)(9,26,27,32,28,30,29,24);(1,4,18,20,7,22,6,21)(2,9,13,27,12,28,11,29)(3,15,16,17,5,19,23,31)(8,24,14,30,25,32,10,26);(1,5)(2,10)(3,7)(4,19)(6,16)(8,13)(9,26)(11,25)(12,14)(15,22)(17,21)(18,23)(20,31)(24,27)(28,30)(29,32)

Polycyclic presentation:

2^9|b^2=e,c^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,f^b=fi,e^c=ei,f^c=fi,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, 330441);