#472869

order	512
nilpotency class	3
isoclinism	isoclinic class of #455699
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2797
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16839
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

822 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)(3,14)(4,9)(5,11)(6,22)(7,12)(8,13)(10,16)(15,25)(17,26)(18,24)(19,28)(20,27)(21,29)(23,31)(30,32);(1,3,8,16)(2,9)(4,7,18,6)(5,15,21,32)(10,22)(11,23)(12,14)(13,24)(17,19,31,20)(25,27)(26,29)(28,30);(1,4,6,16,8,18,7,3)(2,10,12,9,13,14,22,24)(5,15,19,31,21,32,20,17)(11,26,27,30,29,23,28,25);(1,5,6,20,8,21,7,19)(2,11,12,28,13,29,22,27)(3,15,4,17,16,32,18,31)(9,23,14,30,24,26,10,25)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 380065);