#478331

order	512
nilpotency class	3
isoclinism	isoclinic class of #476613
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

797 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)(5,6,12,16,11,17,10,28)(7,18,21,32,22,31,20,13)(8,23,26,15,9,27,25,14)(19,29,35,33,24,30,34,36);(5,7,10,20,11,22,12,21)(6,13,16,18,17,32,28,31)(8,24,25,35,9,19,26,34)(14,29,23,33,15,30,27,36);(5,8,12,26,11,9,10,25)(6,14,28,27,17,15,16,23)(7,19,21,35,22,24,20,34)(13,29,31,36,32,30,18,33);(2,4)(5,9,11,8)(6,15,17,14)(7,19,22,24)(10,25,12,26)(13,30,32,29)(16,23,28,27)(18,33,31,36)(20,35,21,34)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385527);