#451917

order	512
nilpotency class	3
isoclinism	isoclinic class of #451917
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16021
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

785 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:

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

Polycyclic presentation:

2^9|a^2=efgh,c^2=e,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^c=gi,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, 359113);