#394385

order	512
nilpotency class	3
isoclinism	isoclinic class of #394382
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2550 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:

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

Polycyclic presentation:

2^9|b^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,h^a=hi,c^b=cg,g^b=gi,e^c=ei,f^c=fi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301581);