#435268

order	512
nilpotency class	3
isoclinism	isoclinic class of #420517
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15521
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

652 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:

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

Polycyclic presentation:

2^9|a^2=g,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,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, 342464);