#462062

order	512
nilpotency class	3
isoclinism	isoclinic class of #462062
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16415
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

784 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 369258);