#433542

order	512
nilpotency class	3
isoclinism	isoclinic class of #433542
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15484
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

809 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)(3,12,14,7)(4,8,17,26)(5,9)(6,10)(11,20)(13,24)(15,23,19,22)(16,21)(18,25,30,27)(28,32)(29,31);(1,3)(2,7)(4,16,17,13)(5,19)(6,14)(8,24,26,21)(9,23)(10,12)(11,22)(15,20)(18,28,30,29)(25,31,27,32);(1,4)(2,8,10,26)(3,13)(5,18)(6,17)(7,21,12,24)(9,25,11,27)(14,16)(15,28)(19,29)(20,30)(22,31,23,32);(2,9)(5,20)(7,22)(8,25)(10,11)(12,23)(15,19)(18,30)(21,31)(24,32)(26,27)(28,29)

Polycyclic presentation:

2^9|a^2=efg,b^2=ef,c^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,f^c=fi,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, 340738);