#465632

order	512
nilpotency class	3
isoclinism	isoclinic class of #465632
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16526
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23439	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1065 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)(3,13)(4,18)(5,10)(6,12)(7,15)(8,16)(9,19)(11,21)(14,24)(17,23)(20,25)(22,28)(26,31)(27,29)(30,32);(1,3,6,16,5,15,21,17)(2,7,12,23,10,13,11,8)(4,14,19,28,20,30,31,29)(9,22,25,32,26,27,18,24);(1,3)(2,8)(4,14)(5,15)(6,17)(7,11)(9,24)(10,23)(12,13)(16,21)(18,22)(19,29)(20,30)(25,27)(26,32)(28,31);(1,4,6,19,5,20,21,31)(2,9,11,18,10,26,12,25)(3,14,16,28,15,30,17,29)(7,22,8,24,13,27,23,32)

Polycyclic presentation:

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

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, 372828);