#147765

order	512
nilpotency class	4
isoclinism	isoclinic class of #13350
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

1668 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)(7,15,18,26)(8,20,22,12)(9,13,24,28)(10,14)(11,17)(16,31,32,29)(19,30)(21,25)(23,34)(27,36)(33,35);(5,7,10,18)(6,11,14,25)(8,21,22,17)(9,19,24,33)(12,26,20,15)(13,16,28,32)(23,31,36,29)(27,35,34,30);(2,4)(5,6)(7,16)(8,12)(9,13)(10,14)(11,19)(15,29)(17,30)(18,32)(20,22)(21,35)(23,27)(24,28)(25,33)(26,31)(34,36)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,h^a=hi,c^b=cf,d^b=di,e^b=ehi,g^b=ghi,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, 54961);