#476436

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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,d^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, 383632);