#476437

order	512
nilpotency class	3
isoclinism	isoclinic class of #476293
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

857 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=i,e^2=i,g^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,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,g^d=gi,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, 383633);