#147805

order	512
nilpotency class	4
isoclinism	isoclinic class of #13360
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×#535	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1071 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fi,g^c=gi,e^d=ei,g^d=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, 55001);