#147756

order	512
nilpotency class	4
isoclinism	isoclinic class of #13353
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator	C₂×#41	C₂²×C₄
Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1656 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,e^b=eh,f^b=fi,g^b=gh,f^c=fi,h^c=hi,e^d=ei,g^d=gi,f^e=fi,g^f=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54952);