#398734

order	512
nilpotency class	3
isoclinism	isoclinic class of #394483
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2472
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14767
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

962 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 305930);