#6431926

order	512
nilpotency class	3
isoclinism	isoclinic class of #23439
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#121
prime radical	C₂²×#575	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4638 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|c^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,f^b=fi,g^c=gi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339122);