#451700

order	512
nilpotency class	3
isoclinism	isoclinic class of #420055
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2476
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16009
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

584 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358896);