#461728

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

827 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)(5,6,10,14,11,16,9,15)(7,17,21,29,22,18,20,12)(8,23,26,13,27,24,25,30)(19,28,34,31,35,36,33,32);(5,7,9,20,11,22,10,21)(6,12,14,17,16,29,15,18)(8,19,25,33,27,35,26,34)(13,28,24,31,30,36,23,32);(5,6)(7,18,22,17)(8,24)(9,15)(10,14)(11,16)(12,20,29,21)(13,25)(19,28,35,36)(23,27)(26,30)(31,33,32,34);(2,4)(5,8,9,25,11,27,10,26)(6,13,14,24,16,30,15,23)(7,19,20,33,22,35,21,34)(12,28,17,31,29,36,18,32)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=ef,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,f^b=fi,d^c=de,e^c=ei,f^c=fi,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, 368924);