#147535

order	512
nilpotency class	3
isoclinism	isoclinic class of #2018
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#121
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1954
prime radical	C₂²×#135	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1014 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)(5,6,10,16,9,15,18,28,11,14,23,31,27,34,36,35)(7,17,8,22,20,29,25,32,21,12,26,13,19,30,24,33);(2,4)(5,7)(6,12)(8,23)(9,20)(10,26)(11,21)(13,31)(14,17)(15,30)(16,22)(18,24)(19,27)(25,36)(28,32)(29,34)(33,35);(5,8)(6,13)(7,18)(9,25)(10,19)(11,26)(12,28)(14,22)(15,33)(16,29)(17,35)(20,23)(21,36)(24,27)(30,31)(32,34)

Polycyclic presentation:

2^9|a^2=g,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,f^b=fi,g^b=gi,f^c=fi,g^c=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54731);