#470936

order	512
nilpotency class	3
isoclinism	isoclinic class of #470936
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16718
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

718 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:

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

Polycyclic presentation:

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

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, 378132);