#448648

order	512
nilpotency class	3
isoclinism	isoclinic class of #418535
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

970 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)(5,6,11,16)(7,17,18,12)(8,21,22,29)(9,13,10,15)(14,23,30,24)(19,33,32,36)(20,26,25,28)(27,34,31,35);(5,7,9,20,11,18,10,25)(6,12,15,26,16,17,13,28)(8,19,23,34,22,32,24,35)(14,27,29,36,30,31,21,33);(6,13)(7,18)(9,10)(12,26)(14,29)(15,16)(17,28)(19,32)(21,30)(23,24)(27,33)(31,36);(2,4)(5,8)(6,14)(7,19,18,32)(9,24)(10,23)(11,22)(12,27,17,31)(13,29)(15,21)(16,30)(20,35,25,34)(26,33,28,36)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=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, 355844);