#470134

order	512
nilpotency class	3
isoclinism	isoclinic class of #454974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

748 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6,10,15,11,16,9,14)(7,17,21,28,20,27,19,12)(8,13,24,31,25,32,23,30)(18,33,22,26,35,29,34,36);(5,7,9,19,11,20,10,21)(6,12,15,17,16,28,14,27)(8,22,24,35,25,34,23,18)(13,29,30,26,32,33,31,36);(5,8)(6,13)(7,18,20,35)(9,23)(10,24)(11,25)(12,26,28,36)(14,30)(15,31)(16,32)(17,29,27,33)(19,34,21,22);(2,4)(6,14,16,15)(8,23,25,24)(12,27,28,17)(18,34,35,22)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=f,d^2=eg,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^b=fi,d^c=de,e^c=ei,h^c=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, 377330);