#499304

order	512
nilpotency class	4
isoclinism	isoclinic class of #13371
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#476
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#476
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

790 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)(7,20,25,35)(8,21,28,24)(9,16)(10,30)(11,32)(12,34)(13,19)(14,22,27,23)(15,26,33,36)(17,31)(18,29);(5,7,10,22,12,24,31,15,13,25,29,23,9,21,11,33)(6,14,17,20,16,26,30,28,19,27,32,35,34,36,18,8);(2,4)(5,8)(6,15)(7,18)(9,20)(10,26)(11,27)(12,35)(13,28)(14,31)(16,22)(17,21)(19,33)(23,34)(24,32)(25,30)(29,36);(5,9,13,12)(6,16,19,34)(7,21,25,24)(8,20,28,35)(10,11,29,31)(14,26,27,36)(15,22,33,23)(17,30,32,18)

Polycyclic presentation:

2^9|a^2=ef,b^2=e,d^2=i,e^2=h,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gh,h^a=hi,c^b=cg,f^b=fi,g^b=gi,e^c=eh,f^c=fh,g^c=ghi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406500);